Surfaces with Central Configuration and Dulac’s Problem for a Three Dimensional Isolated Hopf Singularity

Abstract

Let $\xi$ be a real analytic vector field with an elementary isolated singularity at $0\in \mathbb {R}^3$ and eigenvalues $\pm bi,c$ with $b,c\in \mathbb {R}$ and $b\ne 0$. We prove that all cycles of $\xi$ in a sufficiently small neighborhood of 0, if they exist, are contained in the union of finitely many subanalytic invariant surfaces, each one entirely composed of a continuum of cycles. In particular, we solve Dulac’s problem for such vector fields, i.e., finiteness of limit cycles.

1 Introduction and Statements

Dulac’s problem is a central topic in the study of the dynamics of real analytic vector fields. In general terms, it consists in proving that there are no infinitely many limit cycles accumulating and collapsing to a singular point. Recall that in general, a cycle (or a closed orbit) of a vector field in a given manifold M is the image of a non-trivial periodic solution $\gamma :\mathbb {R}\rightarrow M$ (also denoted by $\gamma$), and a limit cycle is a cycle possessing a neighborhood free of other cycles.

In dimension two, the problem was answered by Dulac in 1923 [9], but his proof had an important gap. It was solved nearly 70 years after by Ilyashenko [15] and Écalle [11], with two independent and different proofs, both very intricate. Recently, alternative proofs in some particular cases have been published, using o-minimal geometry [8, 12, 17, 30]. Dulac’s result can be used to prove the finiteness statement of (the second part of) Hilbert’s 16th problem. Namely, any polynomial vector field in $\mathbb {R}^2$ has finitely many limit cycles (see Ilyashenko’s survey [16] for more information).

Non-accumulation of limit cycles for planar analytic vector fields implies a stronger property: either there are none in a neighborhood, or there is a continuum family of nested cycles filling a whole punctured neighborhood (a central configuration). In fact, given a cycle $\gamma$, we can define the Poincaré first return map in a transversal segment at some point $p\in \gamma$. It is an analytic local diffeomorphism whose fixed points correspond exactly to cycles in a neighborhood of $\gamma$. Thus, there are necessarily finitely many of them or they form a continuum annulus around $\gamma$. This is also the argument for proving Dulac’s result in the easiest case in dimension two (apart, of course, from the trivial hyperbolic or semi-hyperbolic situations, when no local cycles exist). Namely, the case where the linear part of the vector field has purely imaginary non-zero eigenvalues (a so-called Hopf singularity) since after a blowing-up centered at zero, the exceptional divisor is a cycle and the Poincaré first return map is an analytic map. Hence, the set of fixed points is an analytic set and it can only be either a finite set or a continuum.

In this paper, we solve Dulac’s problem for analytic three-dimensional vector fields with isolated singularity with a pair of conjugated imaginary non-zero eigenvalues (a three-dimensional Hopf singularity). In fact, we determine a finite number of invariant surfaces where local cycles may be placed and theses surfaces present a central configuration. Let us provide precise statements.

Denote by $\mathfrak {X}^{\omega }(\mathbb {R}^3,0)$ the family of germs of analytic vector fields $\xi$ at $0\in \mathbb {R}^3$, singular at the origin, that is, $\xi (0)=0$. If $\xi \in \mathfrak {X}^{\omega }(\mathbb {R}^3,0)$ and U is an open neighborhood of 0 where (a representative of) $\xi$ is defined, we denote by $\mathcal {C}_U=\mathcal {C}_U(\xi )$ the union of all cycles of $\xi |_U$ (that is, entirely contained in U). It is called the cycle-locus of $\xi$ in U. Notice that this cycle-locus depends strongly on the neighborhood U and that it does not behave as a germ of a set that we can associate to the germ $\xi$ (i.e., if $U'\subset U$ we can only assert that $\mathcal {C}_{U'}\subset \mathcal {C}_U$, but not $\mathcal {C}_{U'}=U'\cap \mathcal {C}_U$).

Consider the following family:

$$\begin{aligned} \mathcal {H}^3:=\{\xi \in \mathfrak {X}^{\omega }(\mathbb {R}^3,0): \ \textrm{Spec}(D\xi (0))=\{\pm bi,c\}, \text{ where } b,c\in \mathbb {R} \text{ and } b\ne 0\}. \end{aligned}$$

Observe that any $\xi \in \mathcal {H}^3$ has a unique formal invariant curve $\widehat{\Omega }=\widehat{\Omega }_\xi$ at 0, which is non-singular and tangent to the eigenspace corresponding to the eigenvalue c. It is called the (formal) rotational axis of $\xi$. When $c\ne 0$ (the semi-hyperbolic case), the rotational axis is convergent and provides an analytic invariant curve, since in this case $\widehat{\Omega }$ coincides with the stable or unstable manifold of $\xi$ (see for instance [7]). On the contrary, when $c=0$ (the completely hyperbolic case or zero-Hopf singularity), the rotational axis $\widehat{\Omega }$ may be convergent or not, although there is always an invariant $\mathcal {C}^\infty$-curve whose Taylor expansion at 0 coincides with $\widehat{\Omega }$. This is a result by Bonckaert and Dumortier in [3] in the case where $\xi$ has an isolated singularity since $\xi$ satisfies the required Łojasiewicz inequality condition in this case). It is trivially true if the singularity is not isolated since, in this case, $\widehat{\Omega }$ coincides with the singular locus $\text {Sing}(\xi )$, an analytic curve. Notice that in the semi-hyperbolic case, $\xi$ has an isolated singularity.

The main result in this paper can be stated as follows.

Theorem 1.1

(Structure of cycle-locus) Let $\xi \in \mathcal {H}^3$ with isolated singularity. Then there is some neighborhood U of $0\in \mathbb {R}^3$, where a representative of $\xi$ is defined, for which exactly one of the following possibilities holds:

• (i) $\mathcal {C}_U(\xi )=\emptyset$.

• (ii) There is a finite non-empty family $\mathcal {S}= \{ S_1,\ldots ,S_r \}$ of connected regular analytic two-dimensional submanifolds of $U{\setminus }\{0\}$, mutually disjoint, invariant for $\xi$, subanalytic sets in U and satisfying $\overline{S_j}\cap U=S_j\cup \{0\}$ for any j, such that, for any element $V\subset U$ in some neighborhood basis at 0, we have

  $$\begin{aligned} \mathcal {C}_{V}(\xi ) = (S_1\cup S_2\cup \cdots \cup S_r)\cap V. \end{aligned}$$

As a consequence, Dulac’s property is true for these vector fields:

Corollary 1.2

Let $\xi \in \mathcal {H}^3$ with an isolated singularity. Then there are not infinitely many limit cycles of $\xi$ accumulating and collapsing to $0\in \mathbb {R}^3$.

Fig. 1

Illustration of case (ii). Each surface has a center configuration

In the second possibility (ii), see Fig. 1, the germs of the surfaces $S_j\in \mathcal {S}$ are uniquely determined. Each of them, in a sufficiently small neighborhood, is composed of a continuum of nested cycles around the singularity, i.e., each $S_j$ is a surface with a central configuration as in the planar case (although $S_j$ could be singular at the origin). Let us call each $S_j\in \mathcal {S}$ a limit central surface, by analogy with the concept of limit cycle. The following example defines two limit central surfaces, both being singular at the origin.

Example 1.3

Consider the following vector field in $\mathcal {H}^3$.

$$\begin{aligned} \xi = (-y-xz^2+x(x^2+y^2))\frac{\partial }{\partial x}+ (x-yz^2+y(x^2+y^2))\frac{\partial }{\partial y}+ (z^3-z(x^2+y^2))\frac{\partial }{\partial z}. \end{aligned}$$

It has isolated singularity. The two half-cones $S_1=\{ (x,y,z): x^2+y^2-z^2=0, z>0 \}$ and $S_2=\{ (x,y,z): x^2+y^2-z^2=0, z<0 \}$ are invariant. The restriction of $\xi$ to any of the surfaces $S_i$ is $\xi |_{S_i}= -y\frac{\partial }{\partial x}+x\frac{\partial }{\partial y}$, which proves that $\xi$ defines a central configuration in $S_i$, for $i=1,2$.

As an application of Theorem 1.1 to this example, one can see that there are not cycles outside $S_1\cup S_2$ in a neighborhood of 0.

The result stated in Theorem 1.1 for the semi-hyperbolic case ($c\ne 0$) has been already proved by Aulbach [1], in a more general situation of n-dimensional analytic vector fields with a pair of purely imaginary non-zero eigenvalues and $n-2$ eigenvalues with non-zero real part. Before Aulbach, the same situation has been considered in the literature by other authors [18, 19, 27, 28], under the assumption that the vector field has a first integral (as in the classical Lyapunov’s result [20]). Using that any center manifold contains every local cycle (see [6]), one obtains that the possibility (ii) can only occur for a unique limit central surface ($r=1$), which coincides with the center manifold $W^c$ of $\xi$ (hence unique, non-singular and analytic in this case).

Vector fields with a Hopf singularity in the completely non-hyperbolic case ($c=0$) have been studied in the literature. For instance, Dumortier in [10] considered such vector fields of class $\mathcal {C}^\infty$ at $0\in \mathbb {R}^3$, satisfying two Łojasiewicz-type inequalities: one for the vector field itself, which implies that 0 is an isolated singularity; and a second one for the infinitesimal generator of the Poincaré first-return map associated to the cycle that appears after the blowing-up of an invariant $\mathcal {C}^\infty$ realization of $\widehat{\Omega }$. He obtains a complete description of the asymptotic behavior of all trajectories in a neighborhood of the origin, as well as a weak topological classification of the vector field. However, those assumptions prevent the existence of any local cycle (that is, one has only the possibility (i) of Theorem 1.1). In our situation where $\xi$ is analytic, Łojasiewicz’s inequality for $\xi$ is equivalent to the property of isolated singularity, but we do not require the second assumption, thus permitting the existence of cycles and hence the possibility (ii).

We should mention that vector fields in $\mathcal {H}^3$ have also been considered in families for different purposes. We can mention Guckenheimer and Holmes [14], where there is a complete description of the bifurcation diagrams for small codimension singularities; Baldoma, Ibáñez and Martínez-Seara [2] where the appearance of certain chaotic behavior, associated to a Shilnikov configuration, is studied; García [13], where, for each $k\in \mathbb {N}$, it is shown the existence of a bound for the number of limit cycles, appearing in certain generic families inside $\mathcal {H}^3$, which make at most k turns around the rotational axis.

Let us summarize the ideas for the proof of Theorem 1.1 and the plan of the article.

In Sect. 2, we propose a simple proof in the semi-hyperbolic case, in spite of the existing references already mentioned for this situation. Our aim, apart for the sake of completeness, is to introduce some of the arguments involved in the proof of the general case, absent in Aulbach’s proof [1] but appearing in Dumortier’s work [10]. Namely, blowing-up techniques and Poincaré first return map along the cycles emerging from the blowing-up.

The rest of the article is devoted to the proof in the completely non-hyperbolic case. We fix a formal normal form ${\hat{\xi }}$ (for instance in the sense of Takens [31]) and a sequence of analytic vector fields $\{ \xi _\ell \}_\ell$ that approximate ${\hat{\xi }}$. The approximation must be understood in terms of jet equalities, that is, $j_\ell (\xi _\ell )=j_\ell ({\hat{\xi }})$. Neither the formal normal form $\hat{\xi }$ nor the sequence $\{ \xi _\ell \}_\ell$ are univocally determined. However, once $\hat{\xi }$ is fixed, we choose the vector fields $\xi _\ell$ to be analytically conjugated to $\xi$. Thus, it is enough to prove Theorem 1.1 for some $\xi _{\ell }$, with $\ell$ large enough.

In Sect. 3, we use blowing-ups to study ${\hat{\xi }}$ and its jets approximations $\xi _\ell$. The rotational axis $\widehat{\Omega }$ is not necessarily convergent and we cannot blow it up (or any realization of it) if we want to preserve analyticity. Starting from the blowing-up of the origin, we define recursively sequences of admissible blowing-ups: a composition of blowing-ups centered at either the infinitely near points of $\widehat{\Omega }$ (characteristic singularities) or invariant closed circles of the corresponding strict transforms of ${\hat{\xi }}$ (characteristic cycles). The main result of this section is a reduction of singularities of the normal form $\hat{\xi }$ adapted to our problem. This process may be understood as a refinement, for this situation, of Panazzolo’s result on reduction of singularities of general three-dimensional analytic vector fields [26] (notice that a Hopf singularity is already in the final elementary situation in the sense of Panazzolo). Essentially, the formal normal form ${\hat{\xi }}$ can be viewed as a vector field of revolution by rotating a planar vector field $\hat{\eta }$; the adapted reduction of singularities corresponds to the reduction of singularities of ${\hat{\eta }}$. In a second part of Sect. 3, we discuss how to apply sequences of admissible blowing-ups to the jet approximations $\xi _{\ell }.$ We find lower bounds for $\ell$ so that certain dynamical properties of ${\hat{\xi }}$, that depend on a finite jet, are inherited by $\xi _\ell$. In particular, the characteristic cycles are actual cycles of the strict transform of $\xi _\ell$.

In Sect. 4, we prove that, after any sequence of admissible blowing-ups, the characteristic cycles and the characteristic singularities are the only possible limit sets of families of cycles of the transform of $\xi _{\ell }$, provided that $\ell$ is large enough. Thus, in order to prove Theorem 1.1, we only search for cycles near the characteristic cycles and characteristic singularities.

In Sect. 5, we study the different local situations appearing after an adapted reduction of singularities $\pi :(M,E)\longrightarrow (\mathbb {R}^3,0)$ of ${\hat{\xi }}$. We have specific monotonic functions along the trajectories of the transformed vector field $\widetilde{\xi }_\ell = \pi ^*\xi _\ell$ in neighborhoods of characteristic singularities or corner-characteristic cycles, preventing the existence of cycles of $\widetilde{\xi }_\ell$ in sufficiently small neighborhoods of them. Around a non-corner characteristic cycle $\gamma$, we work with the associated Poincaré first return map $P_\gamma$ of $\widetilde{\xi }_\ell$. First, we find a formal invariant non-singular surface $S_\gamma$ of $\widetilde{\xi }_\ell$ supported by $\gamma$ and transversal to the divisor, using that this is the case for the transform $\pi ^*{\hat{\xi }}$ of ${\hat{\xi }}$. This surface $S_\gamma$ provides a formal invariant curve $\Gamma _\gamma$ for $P_\gamma$ and, around $\Gamma _\gamma$, we can describe the periodic orbits of $P_\gamma$. Namely, there is a conic neighborhood $\Sigma _\gamma$ around $\Gamma _\gamma$ such that: if $\Gamma _\gamma \nsubseteq \text {Fix}(P_\gamma )$, there are not periodic points of $P_\gamma$ inside $\Sigma _\gamma$; if, otherwise, $\Gamma _\gamma \subseteq \text {Fix}(P_\gamma )$, then $\Gamma _\gamma$ is exactly the set of periodic points (thus fixed) inside $\Sigma _\gamma$.

Finally in Sect. 6, we give the proof of Theorem 1.1 gathering the results of the previous sections. First, we fix a vector field $\xi _{\ell }$ to which the reduction of singularities $\pi$ can be applied. By means of the results in Sects. 4 and 5, cycles of ${\widetilde{\xi }}_{\ell }$ sufficiently near to (but not contained in) the divisor E can only be located in neighborhoods of the non-corner characteristic cycles $\gamma$. Moreover, the conic neighborhoods $\Sigma _\gamma$ above provide solid conic neighborhoods ${\widetilde{\Sigma }}_\gamma$ of $S_\gamma$ in such a way that if a cycle of $\widetilde{\xi }_\ell$ is contained in ${\widetilde{\Sigma }}_\gamma$, then the curve $\Gamma _\gamma$ is contained in $\text {Fix}(P_\gamma )$ and supports a continuum of cycles inside the saturation of $\Gamma _\gamma$ by the flow, an analytic surface around $\gamma$. The projection of this surface under $\pi$ provides a limit central surface. This would finish the proof of Theorem 1.1 if we could guarantee that all cycles of $\widetilde{\xi }_\ell$ in a neighborhood of $\gamma$ are contained in the cone ${\widetilde{\Sigma }}_\gamma$. This is achieved by “opening” the cones $\Sigma _\gamma$ to actual neighborhoods of $\gamma$ by means of further blowing-ups. In this way, it is possible that we need a larger jet approximation $\xi _{\ell '}$ with $\ell '\ge \ell$, for which the order of its cones could change, a priori. We overcome this last difficulty showing that the order of a cone around $\gamma$ where the cycles have the desired properties may be uniformly bounded for $\ell '\ge \ell$.

1.1 Notation and Conventions About the Power Series

If A is a $\mathbb {R}-$algebra and $\textbf{x}=(x_1,\ldots ,x_n)$ are variables, $A[[\textbf{x}]]$ denotes the $\mathbb {R}-$algebra of formal power series in $\textbf{x}$ with coefficients in A. Elements $f\in A[[\textbf{x}]]$ are written as

$$\begin{aligned} f=\displaystyle \sum _{\alpha \in \mathbb {N}_{\ge 0}^{n}}f_\alpha \textbf{x}^\alpha , \text {where }\textbf{x}^\alpha :=x_1^{\alpha _1}\cdots x_n^{\alpha _n}\text { if }\alpha =(\alpha _1,\ldots , \alpha _n). \end{aligned}$$

For any $k\ge 0$, the $k-$jet of f is defined as

$$\begin{aligned} j_k(f)=j_k^{\textbf{x}}(f):= \displaystyle \sum _{\alpha : |\alpha |\le k} f_\alpha \textbf{x}^\alpha \end{aligned}$$

where $|\alpha |:=\alpha _1+\cdots +\alpha _n$. The order of f, denoted by $\nu (f)$ is the first $k\ge 0$ (or $+\infty$ if it does not exist) such that $j_k(f)\ne 0$. If we separate the variables into two groups $\textbf{x}=(\textbf{y},\textbf{z})$ where $\textbf{y}=(y_1, \ldots , y_r)$ and $\textbf{z}=(z_1, \ldots , z_s)$, the $k-$jet $j_{k}^{\textbf{z}}(f)$ of f with respect to the variable $\textbf{z}$ is the $k-$jet of f as an element of $A[[\textbf{y}]][[\textbf{z}]]$ under the natural identification $A[[\textbf{x}]]{\mathop {\rightarrow }\limits ^{\sim }} A[[\textbf{y}]][[\textbf{z}]]$, that is, the jet $j_{k}^{\textbf{z}}(f)$ is given by

$$\begin{aligned} j_{k}^{\textbf{z}}(f):= \sum _{|\beta | \le k }\left( \sum _{\gamma \in \mathbb {N}_{\ge 0}^{r}} f_{(\gamma ,\beta )} \textbf{y}^\gamma \right) \textbf{z}^\beta . \end{aligned}$$

We will use freely the following basic properties of jets:

• $j_k(f\cdot g)= j_k( j_k(f) \cdot j_k(g) )$, for $f,g\in A[[\textbf{x}]]$. In fact, this property can be refined: if $k\ge \max \{ \nu (f), \nu (g) \},$ then $j_k(f\cdot g)= j_k (j_{k-\nu (g)}(f) \cdot j_{k-\nu (f)}(g))$.

• $j_k(f^{-1})= j_k( (j_k (f))^{-1} )$ if f is a unit in $A[[\textbf{x}]]$.

• $j_k^{\textbf{z}}(f(x_1,\ldots , x_{i}+a, \ldots , x_n))= j_k^{\textbf{z}} (f)(x_1,\ldots , x_{i}+a, \ldots , x_n)$ for $i\le r$ and $a\in A$.

• $j_k(f)=j_k( j_k^{\textbf{z}}(f))$.

We extend the use of $k-$jets (respectively with respect to $\textbf{z}$) for formal vector fields ${\hat{\eta }}= \eta _1 \frac{\partial }{\partial x_1}+ \cdots + \eta _n \frac{\partial }{\partial x_n}$ with $\eta _j\in A[[\textbf{x}]]$ or tuples $F=(f_1,\ldots ,f_m) \in A [[\textbf{x}]]^m$ in the obvious way

$$\begin{aligned} j_k^u({\hat{\eta }}):= j_k^u(\eta _1) \frac{\partial }{\partial x_1}+ \cdots + j_k^u(\eta _n) \frac{\partial }{\partial x_n},\ \quad j_k^u(F):=(j_k^u(f_1),\ldots , j_k^u(f_m)), \end{aligned}$$

with $u=\textbf{x}$ (respectively $u=\textbf{z}$).

When A is a normed space, the subalgebra of convergent series with coefficients on A is the subalgebra of $A[[\textbf{x}]]$ defined by

$$\begin{aligned} A\{\textbf{x}\}:= \displaystyle \bigcup _{\delta >0} A\{ \textbf{x}\}_\delta \end{aligned}$$

where, by definition, a series $f= \sum _{\alpha \in \mathbb {N}_{\ge 0}^{n}}f_\alpha \textbf{x}^\alpha \in A[[\textbf{x}]]$ belongs to $A\{\textbf{x}\}_\delta$ if there exists $C>0$ such that $||f_\alpha ||<C\delta ^{|\alpha |}$ for any $\alpha$. The main examples for the algebra of the coefficients used along the article are the following:

• $A=\mathbb {R}$ with the standard norm of the absolute value.

• $A=\mathbb {R}[\cos \theta , \sin \theta ]$, the algebra of trigonometric polynomials, whose elements are considered indistinctively as a function on $\mathbb {R}$ or on $\mathbb {S}^1$, via the covering $\tau :\theta \rightarrow (\cos \theta , \sin \theta )$. It will be endowed with the supremum norm $||f||:=\sup _{\theta \in \mathbb {R}}f(\theta )$. Notice that given a convergent series $F\in \mathbb {R}[\cos \theta , \sin \theta ]\{ \textbf{x}\}_\delta$, its partial sums converge absolutely and uniformly in the compact sets of the neighborhood $V=\mathbb {S}^1\times (-\delta ,\delta )^n$ of $\mathbb {S}^1\times \{0\}$ (or the neighborhood $V=\mathbb {R}\times (-\delta ,\delta )^n$ of $\mathbb {R}\times \{0\}$), thus providing an analytic function that we denote again f.

• In the case of $A=\mathbb {R}[\textbf{z}]$ (respectively $\mathbb {R}[\cos \theta , \sin \theta ,\textbf{z}]$), where $\textbf{z}=(z_1,\ldots , z_r)$, there is no unique natural norm on A. We will consider a norm for each compact set K of $\mathbb {R}^r$ (resp. $\mathbb {S}^1\times \mathbb {R}^r$) with non-empty interior, defined by

  $$\begin{aligned} || f||_K:=\sup _{a\in K} \{|f(a)|\}. \end{aligned}$$

  Denoting $A_K=(A,|| \cdot ||_K)$ such a normed space, we have the corresponding algebra of convergent series $A_K\{\textbf{x}\}$. We define the algebra of convergent series with coefficients in A as the intersection of algebras $A_K\{ \textbf{x}\}$ where K runs all compact sets of such form. With an abuse of notation, we name this algebra $A\{\textbf{x}\}$ for convenience. Each element $f\in A\{\textbf{x}\}$ defines an analytic function on a neighborhood of $\mathbb {R}^r\times \{0\}$ (resp. $\mathbb {S}^1\times \mathbb {R}^r\times \{ 0\}$) in $\mathbb {R}^r\times \mathbb {R}^n$ (resp. in $\mathbb {S}^1\times \mathbb {R}^r\times \mathbb {R}^n$).

Moreover, for a formal vector field ${\hat{\xi }}$, we will use the expression $\hat{\xi }(z)$ to denote the formal series obtained by the application of the formal derivation $\hat{\xi }$ to the function z, and it coincides with the coefficient of $\frac{\partial }{\partial z}$ in ${\hat{\xi }}$.

2 The Semi-hyperbolic Case

In this section, we provide a proof of Theorem 1.1 in the semi-hyperbolic case, i.e., the linear part $D\xi (0)$ has eigenvalues $\{\pm bi,c\}$ with both b, c different from zero.

Assume for instance that $c<0$. Then, the stable manifold $W^s$ of $\xi$ at 0 is one-dimensional and, as we have said, it coincides with the rotational axis, which is therefore convergent. Fix some center manifold $W^c$ of $\xi$ at 0 of class $\mathcal {C}^k$, with $k\ge 2$. In general, it is not analytic, nor unique. But it contains any cycle of $\xi$ that is contained in a sufficiently small neighborhood U of the origin, i.e., $\mathcal {C}_U(\xi )\subset W^c$ (see [6]).

Take a neighborhood $U_0$ inside which, both the stable manifold $W^s$ and the chosen center manifold $W^c$ are regular embedded submanifolds, and such that $\mathcal {C}_{U_0}(\xi )\subset W^c$. Let $\pi :M\rightarrow U_0$ be the polar blowing-up with center $W^s$. It is a proper analytic map. The divisor $E=\pi ^{-1}(W^s)$ is a cylinder and the fiber $\gamma =\pi ^{-1}(0)$ over the origin is a cycle of the transformed vector field ${{\widetilde{\xi }}}:=\pi ^*\xi$. The strict transform ${{\widetilde{W^c}}}=\overline{\pi ^{-1}(W^c{\setminus }\{0\})}$ is a surface of class $\mathcal {C}^{k-1}$, invariant for ${{\widetilde{\xi }}}$ and transversal to E. Moreover, $\gamma =E\cap {{\widetilde{W^c}}}$.

Now, consider a point $a\in \gamma$, and two nested analytic discs $\Delta '\subset \Delta$ transverse to ${{\widetilde{\xi }}}$ close to a so that the Poincaré first-return map $P_\gamma :\Delta '\rightarrow \Delta$ of ${{\widetilde{\xi }}}$ associated to $\gamma$ is well defined and analytic. Notice that if $\zeta$ is any cycle of ${{\widetilde{\xi }}}$ such that $\zeta \cap \Delta =\zeta \cap \Delta '$, then the intersection $\zeta \cap \Delta$ is a periodic orbit of $P_\gamma$ (see Fig. 2). In particular, if $\zeta$ is the inverse image by $\pi$ of a cycle inside $\mathcal {C}_{U_0}({{\widetilde{\xi }}})$, then, $\zeta$ is contained in ${{\widetilde{W^c}}}$. Taking into account that $W^c$ is two-dimensional and using classical arguments based on the Jordan Curve Theorem (see for instance [25]), we conclude in this case that $\zeta$ cuts $\Delta '$ in a single point, necessarily a fixed point of $P_\gamma$. Hence, the family of cycles of $\xi$ in a given neighborhood of the origin are in bijection with the set of fixed points $\text {Fix}(P_\gamma )$ of $P_\gamma$ not in E, and hence, $\text {Fix}(P_\gamma )$ is contained in the intersection $H={{\widetilde{W^c}}}\cap \Delta '$. Let us prove now Theorem 1.1.

Fig. 2

Definition of the Poincaré map $P_\gamma$

Suppose that item (i) does not hold, i.e., $\mathcal {C}_U( \xi )\ne \emptyset$ for any open neighborhood U of 0. Then we have infinitely many cycles of $\xi$ that accumulate and collapse to 0. By the above, there are infinitely many fixed points of $P_\gamma$ in H accumulating to the point a. Being $P_\gamma$ an analytic map, its set $\text {Fix}(P_\gamma )$ of fixed points is an analytic set of positive dimension. Since $\text {Fix}(P_\gamma )\subset H$ and H is a curve of class $\mathcal {C}^{k-1}$ (transversal intersection of ${{\widetilde{W^c}}}$ and $\Delta '$), we conclude that $H=\text {Fix}(P_\gamma )$.

Let ${{\widetilde{U}}}$ be a neighborhood of $\gamma$ in M satisfying:

• ${\widetilde{U}}\cap \Delta =\Delta '$.

• ${\widetilde{U}}\cap {\widetilde{W}}^c$ is the saturation of $H\cap {{\widetilde{U}}}$ by the flow of ${{\widetilde{\xi }}}$.

• $U=\pi ({{\widetilde{U}}})$ is contained in $U_0$.

We get that U is a neighborhood of 0 and $\mathcal {C}_{U}(\xi )=W^c\cap U{\setminus }\{0\}$. Notice also that ${{\widetilde{W^c}}}\cap {{\widetilde{U}}}$ is an analytic set since H is an analytic curve. Since $\pi$ is proper, we conclude that $W^c\cap U$ is a subanalytic set and Theorem 1.1 is proved.

Remark 2.1

The proof above shows that, in the semi-hyperbolic case, there is at most one limit central surface $S_1$. Moreover, if $S_1$ exists, then $\overline{S_1}=W^c$ is a center manifold which is unique and analytic (using Tamm’s Theorem [32], because $W^c$ is of class $\mathcal {C}^k$ and subanalytic in this case).

3 Admissible Blowing-Ups and Adapted Reduction of Singularities

Consider a vector field $\xi$ in the family $\mathcal {H}^3$ with completely non-hyperbolic linear part, that is, $\text {Spec}(\xi )=\{ \pm bi,0 \}$. Without loss of generality for the study of the foliation generated by $\xi$, we will assume $b=1.$ In some coordinates, the vector field is written as

$$\begin{aligned} \xi = \left( -y+ A_1(x,y,z) \right) \frac{\partial }{\partial x}+ \left( x+ A_2(x,y,z) \right) \frac{\partial }{\partial y}+ \left( A_3(x,y,z) \right) \frac{\partial }{\partial z}, \end{aligned}$$

(1)

with $A_1,A_2,A_3 \in \mathbb {R}\{x,y,z\}$ of order at least two.

3.1 Formal Normal Form and Truncated Normal Forms

Using Takens’ theorem on normal forms (see [31]), there exists a formal automorphism at 0, expressed in terms of the chosen coordinates as

$$\begin{aligned} \hat{\varphi }(x,y,z)=(x+\hat{\varphi }_1(x,y,z),y+\hat{\varphi }_2(x,y,z), z+\hat{\varphi }_3(x,y,z)) \in \mathbb {R}[[x,y,z]]^3, \end{aligned}$$

with $j_1({\hat{\varphi }}_j)=0$ for $j=1,2,3$, such that the formal vector field $\hat{\xi }=\hat{\varphi }^*(\xi )$ is written in the form

$$\begin{aligned} \hat{\xi }= & T(x^2+y^2,z) \left( -y\frac{\partial }{\partial x}+ x\frac{\partial }{\partial y}\right) + R(x^2+y^2,z) \left( x\frac{\partial }{\partial x}+y \frac{\partial }{\partial y}\right) \nonumber \\ & \quad + Z(x^2+y^2,z)\frac{\partial }{\partial z}, \end{aligned}$$

(2)

where $R,T,Z\in \mathbb {R}[[u,v]]$ and $T(0,0)=1$. Note that $R(u,v), Z(u,v) \in (u,v)$ and $Z(0,v) \in (v^2)$. Remark also that neither the automorphism $\hat{\varphi }$ need to be convergent, nor the components of $\hat{\xi }$ need to belong to $\mathbb {R}\{ x,y, z \}$. Any formal vector field ${\hat{\xi }}$ as in (2) obtained as above is called a formal normal form of $\xi$. We remark that ${\hat{\xi }}$ is not uniquely determined by $\xi$.

Remark 3.1

The z-axis is sent to the formal rotational axis $\widehat{\Omega }$ of $\xi$ by ${\hat{\varphi }}$, that is, $\widehat{\Omega }= {\hat{\varphi }} (0,0,z)$. On the other hand, since ${\hat{\varphi }}$ must preserve the (formal) singular locus, the hypothesis that $\xi$ has isolated singularity implies that $Z(0,v)\ne 0$.

Once we fix a formal normal form ${\hat{\xi }}$ of $\xi$ given by ${\hat{\xi }}= {\hat{\varphi }}^*\xi$, we can consider truncated normal forms of $\xi$ in the following way. For any $\ell \in \mathbb {N}_{\ge 2}$, let $\varphi _\ell$ be the polynomial tangent to the identity diffeomorphism of $(\mathbb {R}^3,0)$ given by

$$\begin{aligned} \varphi _\ell (x,y,z)=(j_{\ell +1} \hat{\varphi })(x,y,z)= (j_{\ell +1}(x\circ {\hat{\varphi }}), j_{\ell +1}(y\circ {\hat{\varphi }}), j_{\ell +1}(z\circ {\hat{\varphi }})). \end{aligned}$$

The vector field $\xi _\ell =(\varphi _\ell )^*(\xi )$ has the same $\ell -$ jet as the formal one $\hat{\xi }$ in coordinates (x, y, z), that is, $j_\ell (\xi _\ell )=j_\ell (\hat{\xi })$. Notice that the vector field $\xi _{\ell }$ is analytically conjugated to $\xi$ and formally conjugated to $\hat{\xi }$ for any $\ell$. More precisely, we have the following formal equation:

$$\begin{aligned} {\hat{\xi }} = \psi _\ell ^*\xi _{\ell },\ \text {where }\psi _\ell := \varphi _\ell ^{-1}\circ \hat{\varphi }. \end{aligned}$$

(3)

It is sufficient to prove Theorem 1.1 for $\xi _\ell$ for any $\ell$. The strategy is the following: we use $\hat{\xi }$ as a guiding vector field so that, after a sequence of blowing-ups, we get a transform of $\hat{\xi }$ with a specific good expression. Both the choice of the sequence of blowing-ups and the expression of the transform will depend only on a finite jet of $\hat{\xi }$, allowing us to choose $\ell$ sufficiently large so that all the construction is applied to $\xi _\ell$.

The blowing-ups will be real (oriented) ones, thus generating boundary and corners, either with center at a point or at an analytic curve isomorphic to the circle $\mathbb {S}^1$. See for instance the work [24] for intrinsic and general definitions of real blowing-ups.

3.2 The First Blowing-Up

The first blowing-up to be done is the real blowing-up $\sigma _0: (M_0,E_0)\longrightarrow (\mathbb {R}^3,0)$ with center at the origin. The blown-up space $M_0$ is a manifold having the divisor $E_0= \sigma _0^{-1}(0)$ as its boundary. This divisor is homeomorphic to a sphere and represents the space of all the half-lines through 0. The morphism $\sigma _0$ defines an analytic isomorphism from $M_0{\setminus } E_0$ to $\mathbb {R}^3{\setminus }\{0 \}$. We consider $M_0$ covered by three charts $(C_0, (\theta ,z^{(0)},\rho ^{(0)}))$, $(C_\infty ,(x^{(\infty )},y^{(\infty )},z^{(\infty )}))$ and $(C_{-\infty },(x^{(-\infty )},y^{(-\infty )},z^{(-\infty )}))$ where $C_0\simeq \mathbb {S}^1 \times \mathbb {R}\times \mathbb {R}_{\ge 0}$ and $C_{\pm \infty }\simeq \mathbb {R}^2\times \mathbb {R}_{\ge 0}$. In these charts, the expression of $\sigma _0$ is given by:

$$\begin{aligned}&\text {In }C_0:&\left\{ \begin{matrix} x & =& \rho ^{(0)} \cos \theta \\ y & =& \rho ^{(0)} \sin \theta \\ z & =& \rho ^{(0)} z^{(0)} \end{matrix} \right. \qquad (\cos \theta , \sin \theta )\in \mathbb {S}^1, z^{(0)}\in \mathbb {R}, \rho ^{(0)} \ge 0 \end{aligned}$$

(4)

$$\begin{aligned}&\text {In }C_\infty :&\left\{ \begin{matrix} x & =& x^{(\infty )} z^{(\infty )}\\ y & =& y^{(\infty )} z^{(\infty )} \\ z & =& z^{(\infty )} \end{matrix} \right. \qquad x^{(\infty )},y^{(\infty )} \in \mathbb {R}, z^{(\infty )}\ge 0 \end{aligned}$$

(5)

$$\begin{aligned}&\text {In }C_{-\infty }:&\left\{ \begin{matrix} x & =& x^{(-\infty )} z^{(-\infty )}\\ y & =& y^{(-\infty )}z^{(-\infty )}\\ z & =& -z^{(-\infty )} \end{matrix} \qquad x^{(-\infty )},y^{(-\infty )}\in \mathbb {R}, z^{(-\infty )}\ge 0. \right. \end{aligned}$$

(6)

Remark 3.2

Strictly speaking, $C_0$ is not the domain of a usual chart of $M_0$, since it is not homeomorphic to an open set of $\mathbb {R}^2\times \mathbb {R}_{\ge 0}$. Considering the usual covering ${\tilde{C}}_0=\mathbb {R}^2\times \mathbb {R}_{\ge 0}$ with $\tau :{\tilde{C}}_0\longrightarrow C_0$ given by $( \theta , z, \rho )\mapsto (\sin \theta , \cos \theta , z, \rho )$, we can treat $\theta$ as a true coordinate (and we will tacitly do), so that $\sigma _0\circ \tau$ has the expression in (4). This convention justifies our abuse of terminology in expressions like “a chart $(C_0,(\theta , z^{(0)},\rho ^{(0)}))$”.

The origins of the charts $C_\infty$ and $C_{-\infty }$ will be denoted by $\gamma _\infty$ and $\gamma _{-\infty }$, respectively. They are the points of the divisor $E_0$ corresponding to the half-lines contained in the $z-$axis and they are the only points of $E_0$ not covered by $C_0$. More explicitly, $\sigma _0(C_0)=\mathbb {R}^3{\setminus } \{ x=y=0 \}$.

We define the (total) transform of ${\hat{\xi }}$ by $\sigma _0$ in the chart $C_0$ as the pull-back

$$\begin{aligned} \hat{\xi } ^{(0)}:= (\sigma _0|_{C_0})^*\hat{\xi }. \end{aligned}$$

Using simplified notation $(z,\rho ):= (z^{(0)},\rho ^{(0)})$ and Eqs. (2) and (4), the vector field $\hat{\xi } ^{(0)}$ is given by

$$\begin{aligned} \hat{\xi } ^{(0)} = B_\theta (z,\rho ) \frac{\partial }{\partial \theta }+ B_z(z,\rho ) \frac{\partial }{\partial z}+ B_\rho (z,\rho ) \frac{\partial }{\partial \rho }, \end{aligned}$$

(7)

where $B_\theta (z,\rho )= T(\rho ^2,\rho z)$, $B_z(z,\rho )= \frac{1}{\rho }Z(\rho ^2,\rho z) -zR(\rho ^2,\rho z)$ and $B_\rho (z,\rho )=\rho R(\rho ^2,\rho z)$. Notice that, by the definition of the blowing-up, we have that $B_\theta , B_z,B_\rho \in \mathbb {R}[z][[\rho ]]$ since z is replaced by $z\rho$. Moreover, $( B_z,B_\rho )\ne (0,0)$ since $Z(u,v)\ne 0$ by Remark 3.1 and $\rho$ divides $B_z,B_\rho$.

The coefficient $B_\theta (z,\rho )$ is a unit in $\mathbb {R}[z][[\rho ]]$ since $B_\theta (z,0)=1$. This allows us to consider $\theta$ as the “time variable" and, consequently, $\hat{\xi }^{(0)}$ is completely described by the associated two dimensional formal vector field $\hat{\eta }_0$ given by the system of formal ODEs

$$\begin{aligned} {\hat{\eta }}_0: \left\{ \begin{array}{lcll} \frac{d z}{d \theta } & =& B_\theta (z,\rho )^{-1}B_z(z,\rho ) & =\rho ^{n^{(0)}} A_z(z,\rho )\\ \frac{d \rho }{d \theta } & = & B_\theta (z,\rho )^{-1}B_\rho (z,\rho ) & =\rho ^{n^{(0)}} A_\rho (z,\rho ). \end{array} \right. \end{aligned}$$

(8)

In this expression, $A_i\in \mathbb {R}[z][[\rho ]]$ for $i=z,\rho$ and ${n^{(0)}}$ is the maximum exponent n such that $\rho ^n$ divides both $B_\rho$ and $B_z$. The associated reduced vector field is by definition $\hat{\eta }'_0:=\rho ^{-n^{(0)}}\hat{\eta }_0$.

There are two possible scenarios determined in the following definition.

Definition 3.3

The blowing-up $\sigma _0$ is called non-dicritical if $A_\rho (z,0)\equiv 0$ and dicritical if $A_\rho (z,0)\ne 0$. Alternatively, we say that $E_0$ is non-dicritical or that $E_0$ is dicritical, respectively.

Despite of the fact that ${\hat{\eta }}_0$ is just formal, the restriction ${\hat{\eta }}_0'|_{F_0}$ to the curve $F_0:=E_0\cap \{\theta =0\}$ is a well defined vector field (under the natural identification $\{ \theta = 0 \} \cong \mathbb {R}^2$, $(0,z,\rho )= (z,\rho )$). This restriction has polynomial coefficients in the coordinate z. Therefore, its singular locus:

$$\begin{aligned} \text {Sing}({\hat{\eta }}_0'|_{F_0}):=\{ a\in F_0:\ {\hat{\eta }}'_0 |_{F_0}(a)=0\}=\{ (z,0):\ A_\rho (z,0)=A_z(z,0)=0 \} \end{aligned}$$

is finite. Singular points are points where we have to focus in order to define successive blowing-ups. But, in the dicritical case, we have to add those non-singular points where the vector field is tangent to the divisor. To be used for later, we recall the definition of such non-transversal points in the general situation of a normal crossing divisor (see Cano, Cerveau and Deserti’s book [5] in the complex holomorphic context).

Let $\chi$ be a formal vector field defined at 0 and F a non-empty normal crossing divisor. Consider a chart (U, (x, y)) centered at 0 where $F=\{ xy^\epsilon =0 \}$ and the coefficients of the vector field in these coordinates belong to $\mathbb {R}[y][[x]]$ if $\epsilon =0$ or to $\mathbb {R}[y][[x]]\cap \mathbb {R}[x][[y]]$ if $\epsilon =1$. Take any point $a=(a_1,a_2)\in F\cap U$, the vector field $\chi _a:=\chi ({\tilde{x}}+ a_1,{\tilde{y}} + a_2)$ is well defined as a formal vector field in coordinates $({\tilde{x}},{\tilde{y}})$.

Definition 3.4

Let F be a non-empty normal crossing divisor and let $\chi$ be a formal vector field defined at F. The adapted singular locus ${{\widetilde{\text {Sing}}}}(\chi ,F)$ of $\chi$ relatively to F, is the set of points $p\in F$ in which either $\chi (p)=0$ or $C\cup F$ has no normal crossings at p, where C is the formal invariant curve of $\chi$ through p.

Applied to our reduced vector field ${\hat{\eta }}_0'$ and to $F_0$, we have

• (a) If $E_0$ is non-dicritical, then ${{\widetilde{\text {Sing}}}}({\hat{\eta }}_0', F_0)=\text {Sing}({\hat{\eta }}_0'|_{F_0})$.

• (b) If $E_0$ is dicritical, then ${{\widetilde{\text {Sing}}}}({\hat{\eta }}_0,F_0)=\text {Sing}({\hat{\eta }}_0'|_{F_0})\cup \{ (z,0): A_\rho (z,0)=0 \}$

In both cases, the adapted singular locus ${{\widetilde{\text {Sing}}}}({\hat{\eta }}_{0}, F_0)$ is finite.

We define also the transforms ${\hat{\xi }}^{(\infty )}:= (\sigma _0|_{C_\infty })^*\hat{\xi }$ and $\hat{\xi }^{(-\infty )}:= (\sigma _0|_{C_{-\infty }})^*\hat{\xi }$ of $\hat{\xi }$ in the charts $C_\infty , C_{-\infty }$, respectively. The expressions for $\hat{\xi } ^{(\infty )}$, using simplified notation $(x,y,z):=(x^{(\infty )},y^{(\infty )},z^{(\infty )})$ is the following:

$$\begin{aligned} \begin{aligned} \hat{\xi }^{(\infty )}&= R^{(\infty )}({x}^2+{y}^2, z) \left( {x}\frac{\partial }{\partial {x}}+ {y}\frac{\partial }{\partial {y}} \right) + T^{(\infty )}({x}^2+{y}^2, z) \left( -{y}\frac{\partial }{\partial {x} }+ {x}\frac{\partial }{\partial {y}} \right) \\&\quad + Z^{(\infty )}({x}^2+{y}^2, z) \frac{\partial }{\partial z}, \end{aligned} \end{aligned}$$

(9)

where $R^{(\infty )},T^{(\infty )},Z^{(\infty )}\in \mathbb {R}[{x}^2+{y}^2][[z]]$ are given by:

$$\begin{aligned} R^{(\infty )}({x}^2+{y}^2, z)= & R(({x}^2+{y}^2)z^2, z)-\frac{1}{z}Z(({x}^2+{y}^2)z^2, z),\\ T^{(\infty )}({x}^2+{y}^2, z)= & T(({x}^2+{y}^2)z^2, z) \text { and }\\ Z^{(\infty )}({x}^2+{y}^2, z)= & Z(({x}^2+{y}^2)z^2, z). \end{aligned}$$

In a similar way, we obtain expressions for $\hat{\xi }^{(-\infty )}$.

3.3 Characteristic Cycles and Successive Blowing-Ups

Recall that the adapted singular locus ${{\widetilde{\text {Sing}}}}(\hat{\eta }_0', F_0)$ of $\hat{\eta }'_0$ relative to $F_0$ is finite. Its elements, belonging to $F_0=\{ \theta =\rho ^{(0)}=0 \}$ are determined by the $z^{(0)}-$coordinate in the chart $C_0$. Denote them by

$$\begin{aligned} {{\widetilde{\text {Sing}}}}(\hat{\eta }_0', F_0)=\{ (\omega _i^{(0)},0): i=1,\ldots ,m_0 \}, \quad \text { with }\omega _i^{(0)}< \omega _j^{(0)}\text { if } i<j. \end{aligned}$$

Definition 3.5

The characteristic cycles of $\hat{\xi }$ in ${M}_0$ are the connected components of the set $\mathbb {S}^1\times {{\widetilde{\text {Sing}}}} (\eta _0'|{F_0}) \subset C_0$, that is, the circles in the divisor $E_0$ given by $\gamma _i:=\{ z^{(0)}= \omega ^{(0)}_i, \rho ^{(0)}=0 \}$ for $i=1,2,\ldots , m_0$. The origins $\gamma _\infty , \gamma _{-\infty }$ of the charts $C_\infty$ and $C_{-\infty }$ (cf. Eqs. (5) and (6)) are called the characteristic singularities of $\hat{\xi }$ in ${M}_0$. We use the term characteristic elements to refer to either the characteristic cycles or characteristic singularities.

In the rest of this section, we inductively define certain sequences of blowing-ups attached to $\hat{\xi }$ starting from the data defined above for the first blowing-up $\sigma _{0}$. More precisely, consider the tuple $\mathcal {M}_0:=(M_0,\sigma _0,\mathcal {A}_0,\mathcal {D}_0)$, where:

• $\mathcal {A}_0$ is the atlas of $M_0$ composed by the charts $C_{-\infty },C_0,C_\infty$,

• $\mathcal {D}_0$ is the family of characteristic elements of $\hat{\xi }$ in $M_0$, that is, $\mathcal {D}_0:=\{ \gamma _{-\infty },\gamma _1, \ldots , \gamma _{m_0},\gamma _\infty \}$.

By definition, we say that $\mathcal {M}_0$ is a sequence of admissible blowing-ups of length $l=0$ for $\hat{\xi }$. Suppose that we have already defined sequences of admissible blowing-ups for $\hat{\xi }$ of length $l-1$, consisting on tuples $\mathcal {M}=(M,\pi ,\mathcal {A}, \mathcal {D})$ satisfying the following hypothesis:

• (H1) $\pi :(M,E)\longrightarrow (\mathbb {R}^3,0)$ is a sequence of (real) blowing-ups with smooth analytic closed centers and factorizing through $\sigma _0$ (i.e., $\pi = \sigma _0 \circ {\bar{\pi }}$, where $\bar{\pi }: M\longrightarrow M_0$ is either the identity or a sequence of blowing-ups with smooth analytic closed centers).

• (H2) $\mathcal {D}= \{ \gamma _I \}_{I\in \mathcal I}$ is a finite family of disjoint closed subsets of the divisor $E= \pi ^{-1}(0)$, such that:

  • There are two elements in $\mathcal {D}$ with indices $I_\infty ^\mathcal {M}= (\infty ,{\mathop {\ldots }\limits ^{s}}, \infty )$ and $I_{-\infty }^\mathcal {M}= (-\infty ,{\mathop {\ldots }\limits ^{t}}, -\infty )$ for some $s,t\in \mathbb {N}_{\ge 1}$, that are the two points where E intersects the strict transform $\pi ^*(\{x=y=0\})$ of the $z-$axis. They are called the characteristic singularities of $\hat{\xi }$ in M.

  • The rest of the elements $\gamma _I$, with $I\ne I_{-\infty }^\mathcal {M}, I_{\infty }^\mathcal {M}$, are analytic embedded circles called characteristic cycles (of $\hat{\xi }$ in M).

  • The intersection of any pair of components of E is an element of $\mathcal {D}.$ Each of them is called a corner characteristic cycle. The corner characteristic cycles are those indexed by tuples $I=(i_1,\ldots , i_r)\ne I_{-\infty }^\mathcal {M}, I_{\infty }^\mathcal {M}$ for which $i_r=\pm \infty .$

• (H3) $\mathcal {A}=\{C_J\}_{J \in \mathcal {J}}$ is an atlas of M with the following properties:

  • (1) There are charts $(C_J,(x^{(J)}, y^{(J)},z^{(J)} ))$ centered at the characteristic singularities $\gamma _J$, with $J\in \{ I_\infty ^\mathcal {M}, I_{-\infty }^\mathcal {M} \}$, satisfying $E\cap C_J= \{ z^{(J)} =0 \}$. Moreover, the expression of $\pi$ in the chart $C_{J}$ with $J=I_{\epsilon \infty }^\mathcal {M}$, for $\epsilon =\pm 1$, is

    $$\begin{aligned} \pi (x^{(J)}, y^{(J)},z^{(J)})=((z^{(J)})^rx^{(J)}, (z^{(J)})^ry^{(J)},\epsilon z^{(J)}) \end{aligned}$$

    with $r\in \mathbb {N}_{\ge 1}$ (r and $\epsilon$ depend on J). Furthermore, the coefficients of $\hat{\xi }^{(J)}:=(\pi |_{C_J})^*\hat{\xi }$ belong to $\mathbb {R}[x ^{(J)},y^{(J)}][[z^{(J)}]]$.

  • (2) If $J\notin \{ I_\infty ^\mathcal {M}, I_{-\infty }^\mathcal {M} \}$, the chart $(C_J,(\theta ,z^{(J)},\rho ^{(J)}))$ is defined for $\theta \in \mathbb R$, $z^{(J)}$ in $\mathbb {R}$ or $\mathbb {R}_{\ge 0}$ and $\rho ^{(J)}\in \mathbb {R}_{\ge 0}$ (with the same convention as in Remark 3.2), and satisfies $E\cap C_J=\{ \rho ^{(J)} (z^{(J)})^\epsilon =0 \}$ with $\epsilon =0$ or 1 according to $z^{(J)}$ being defined either in $\mathbb {R}$ or $\mathbb {R}_{\ge 0}$, respectively. In the case $\epsilon =0$, the chart $C_J$ is a non-corner chart and the characteristic cycles contained in $E\cap C_J$ are given by equations $\{ z^{(J)}=a_i, \rho ^{(J)}=0 \}$, where $\{ a_i \}_i$ is a finite collection of real numbers. In the case $\epsilon =1$, the chart $C_J$ is a corner chart and the family of characteristic cycles contained in $E\cap C_J$ consists of a unique corner characteristic cycle given by $\{ z^{(J)}=0, \rho ^{(J)}=0 \}$ and a collection of non-corner characteristic cycles given either by $\{ z^{(J)}=b_j, \rho ^{(J)}=0 \}_j$ for a family $\{b_j\}_j$ of positive numbers or by $\{ z^{(J)}=0, \rho ^{(J)}=c_k \}_k$ for a family $\{c_k\}_k$ of positive numbers.

  • (3) For any $J\notin \{ I_\infty ^\mathcal {M}, I_{-\infty }^\mathcal {M} \}$ the expression of $\pi |_{C_J}$ is polynomial in $(\cos \theta , \sin \theta , z^{(J)},\rho ^{(J)})$ and the transformed vector field $\hat{\xi }^{(J)}:=(\pi |_ {C_J})^*\hat{\xi }$ written, with simplified notation $(\theta ,z,\rho )=(\theta , z^{(J)}, \rho ^{(J)})$, as

    $$\begin{aligned} \hat{\xi }^{(J)}= B_\theta ^{(J)}(z,\rho ) \frac{\partial }{\partial \theta }+ B_z ^{(J)}(z,\rho ) \frac{\partial }{\partial z}+ B ^{(J)}_\rho (z,\rho ) \frac{\partial }{\partial \rho }, \end{aligned}$$

    (10)

    satisfies that, for $i=\theta ,z,\rho$, the coefficient $B_i^{(J)}$ belongs to $\mathbb {R}[z][[\rho ]]$ if $C_J$ is a non-corner chart, or to both algebras $\mathbb {R}[z][[\rho ]]$ and $\mathbb {R}[\rho ][[z]]$, if $C_J$ is a corner characteristic chart. In any case, $B_\theta ^{(J)}(z,0)=1$ and hence it is a unit of the corresponding algebra.

When $J=I_{-\infty }^\mathcal {M}$ or $J=I_{\infty }^\mathcal {M}$, we define $n^{(J)}$ to be the maximum n such that ${\hat{\xi }}^{(J)}(z^{(J)})=(z^{(J)})^{n}\cdot {\tilde{B}} (x^{(J)}, y^{(J)},z^{(J)})$ with ${\tilde{B}}$ an element in $\mathbb {R}[x^{(J)},y^{(J)}][[z^{(J)}]]$.

Observing (H3)-(2), for $J\notin \{ I_\infty ^\mathcal {M}, I_{-\infty }^\mathcal {M} \}$, we define the vector field associated to the transform $\hat{\xi }^{(J)}$, as the formal two dimensional vector field $\hat{\eta }_J$ given by the following system of ODEs (using (10) and simplifying $(z,\rho )=(z^{(J)},\rho ^{(J)})$):

$$\begin{aligned} \hat{\eta }_J: \left\{ \begin{array}{lcll} \frac{d z}{d \theta } & =& B_\theta ^{(J)}(z,\rho )^{-1}B_z^{(J)}(z,\rho ) & =\rho ^{n_1^{(J)}} z^{n_2^{(J)}} A_z^{(J)}(z,\rho )\\ \frac{d \rho }{d \theta } & = & B_\theta ^{(J)}(z,\rho )^{-1}B_\rho ^{(J)}(z,\rho ) & =\rho ^{n_1^{(J)}} z^{n_2^{(J)}} A_\rho ^{(J)}(z,\rho ). \end{array} \right. \end{aligned}$$

(11)

Here, $n^{(J)}_1$ is the maximum n such that $\rho ^n$ divides both $B_\rho ^{(J)}$ and $B_z^{(J)}$ (and thus, $A_\rho ^{(J)}$ and $A_z^{(J)}$ are formal series not both together divisible by $\rho$). On the other hand, if $C_J$ is a non-corner chart, we take $n_2^{(J)}=0$, and, if $C_J$ is a corner chart, we take $n_2^{(J)}$ to be the maximum m such that $z^m$ divides both $B_\rho ^{(J)}$ and $B_z^{(J)}$. We define $n ^{(J)}:=\max \{ n^{(J)}_1, n^{(J)}_2 \}$ in all cases.

It is clear that $\mathcal M_0$ fulfills (H1-H3). Now, a sequence of admissible blowing-ups for $\hat{\xi }$ of length l is a tuple $\mathcal {M'}:=(M',\pi ',\mathcal {A}', \mathcal {D}')$ built from a sequence of admissible blowing-ups $\mathcal {M}=(M,\pi ,\mathcal {A}, \mathcal {D})$ of length $l-1$ in such a way that $\pi '= \pi \circ \sigma _{\gamma _{I}}$, where

$$\begin{aligned} \sigma _{\gamma _I}: M'\longrightarrow M \end{aligned}$$

is the blowing-up centered at some $\gamma _I\in \mathcal {D}$. The expression of $\sigma _{\gamma _{I}}$ in charts and the description of the families $\mathcal {D}',\mathcal {A}'$ are exposed in what follows (see Fig. 3 for an illustration of the different situations). We consider two cases: the blowing-up $\sigma _{\gamma _{I}}$ is centered at a characteristic singularity or at a characteristic cycle.

Fig. 3

Several sequences of admissible blowing-ups

(i) First case. The blowing-up $\sigma _{\gamma _I}$ is centered at the singular point $\gamma _{I}$ with $I=I_\infty ^\mathcal {M}$ (or analogously for $I=I_{-\infty }^\mathcal {M}$). Put $J=I$, $J_\infty =(J,\infty )$ and $J_{0}=(J,0)$. The point $\gamma _{I}$ is the origin of a chart $(C_{J},(x^{(J)}, y^{(J)}, z^{(J)}))\in \mathcal {A}$ where the divisor $E=\pi ^{-1}(0)$ is given by $\{ z^{(J)}=0 \}$. Then, the exceptional divisor $\sigma _{\gamma _I}^{-1}(\gamma _I)$ is covered by two charts of $M'$, say $(C_{J_\infty },(x^{(J_\infty )}, y^{(J_\infty )}, z^{(J_\infty )}))$ and $(C_{J_0}, ( \theta , z^{(J_0)},\rho ^{(J_0)}))$, so that $\sigma _{\gamma _I}$ is written as:

$$\begin{aligned}&\text {In }C_{J_0}:&\left\{ \begin{matrix} x^{(J)} & =& \rho ^{(J_0)} \cos \theta \\ y^{(J)} & =& \rho ^{(J_0)} \sin \theta \\ z^{(J)} & =& \rho ^{(J_0)}z^{(J_0)} \end{matrix} \right. \qquad \theta \in \mathbb {R}, z^{(J_0)},\rho ^{(J_0)} \ge 0 \end{aligned}$$

(12)

$$\begin{aligned}&\text {In }C_{J_\infty }:&\left\{ \begin{matrix} x ^{(J)}& =& x ^{(J_\infty )} z ^{(J_\infty )}\\ y^{(J)} & =& y ^{(J_\infty )} z ^{(J_\infty )} \\ z ^{(J)} & =& z ^{(J_\infty )} \end{matrix} \right. \qquad x ^{(J_\infty )},y ^{(J_\infty )} \in \mathbb {R}, z ^{(J_\infty )} \ge 0. \end{aligned}$$

(13)

We set the atlas of $\mathcal M'$ to be $\mathcal {A}'=(\mathcal {A} {\setminus } \{ C_J \}) \cup \{ C_{J_0}, C_{J_\infty } \}$ (under the identification $\sigma _{\gamma _J}:M'{\setminus } \sigma _{\gamma _J} ^{-1}{(\gamma _J)}\rightarrow M{\setminus } {\gamma _J}$). The chart $C_{J_0}$ is a corner chart in this case.

Let us define $\mathcal {D}$. We consider first the vector field $\hat{\xi } ^{(J_\infty )}:=(\sigma _ {\gamma _I}|_{C_{J_\infty }})^*\hat{\xi } ^ {(I)}$. Define $n ^{(J_\infty )}$ as the maximum $n\in \mathbb N$ such that $(z ^{(J_\infty )})^{n}$ divides $\hat{\xi } ^{(J_\infty )}(z^{(J_\infty )}).$ In the chart $C_{J_\infty }$, the expression of the vector field $\hat{\xi }^{(J_{\infty })}$ is similar to (9). The origin of $C_{J_\infty }$ is named $\gamma _{J_\infty }$. Secondly, consider the formal vector field $\hat{\xi }^{(J_0)}:=(\sigma _{\gamma _I}|_{C_{J_0}})^*(\hat{\xi }^{(I)})$. Use the expression in (10) for $\hat{\xi }^{(I)}$, and rename $(z,\rho )=(z^{(J_0)},\rho ^{(J_0)})$. Then, $\hat{\xi }^{(J_0)}$ is given by

$$\begin{aligned} \hat{\xi }^{(J_0)}= B_\theta ^{(J_0)}(z,\rho ) \frac{\partial }{\partial \theta }+ B_z ^{(J_0)}(z,\rho ) \frac{\partial }{\partial z}+ B ^{(J_0)}_\rho (z,\rho ) \frac{\partial }{\partial \rho }, \end{aligned}$$

(14)

where $B_\theta ^{(J_0)}\in \mathbb {R}[z][[\rho ]]$ is a unit in the algebra $\mathbb {R}[z][[\rho ]]$ (because $B^{(J_0)}_\theta (z,0)=1$) and

$$\begin{aligned} B_z^{(J_0)}=\rho ^{n_1^{(J_0)}} z^{n_2^{(J_0)}} \tilde{B}_z^{(J_0)}, \quad B ^{(J_0)}_\rho =\rho ^{n_1^{(J_0)}} z^{n_2^{(J_0)}} \tilde{B} ^{(J_0)}_\rho , \qquad \tilde{B}_\rho ^{(J_0)},\tilde{B}_z ^{(J_0)}\in \mathbb {R}[z][[\rho ]], \end{aligned}$$

(15)

where $n_1^{(J_0)}, n_2^{(J_0)}$ are defined similarly as we have defined $n_1^{(J)}$ and $n_2^{(J)}$. The vector field associated to $\hat{\xi }^{(J_0)}$ is the two dimensional vector field $\hat{\eta }_{J_0}$ with coefficients in $\mathbb {R}[z][[\rho ]]$, defined in a similar manner as $\hat{\eta }_0$ in (8). That is,

$$\begin{aligned} \hat{\eta }_{J_0}= \rho ^{n_1^{(J_0)}} z^{n_2^{(J_0)}} \left( A_z^{(J_0)}(z,\rho ) \frac{\partial }{\partial z}+ A_\rho ^{(J_0)}(z,\rho )\frac{\partial }{\partial \rho }\right) , \end{aligned}$$

(16)

where $A_k^{(J_0)}={\tilde{B}}_k^{(J_0)}\cdot (B_\theta ^{(J_0)})^{-1}$ for $k=z,\rho .$ The vector field $\hat{\eta }_{J_0}':=\frac{1}{\rho ^{n_1^{(J_0)}} z^{n_2^{(J_0)}}}\hat{\eta }_{J_0}$ is called the reduced vector field associated to $\hat{\xi }^{(J_0)}$. We distinguish two cases:

• The blowing-up $\sigma _{\gamma _I}$ is non-dicritical if $A_\rho ^{(J_0)}(z,0)\equiv 0$. In this case, we say that the divisor $\sigma _{\gamma _I}^{-1}(\gamma _I)$ is a non-dicritical component of the total divisor $E':=(\pi ')^{-1}(0)$.

• The blowing-up $\sigma _{\gamma _I}$ is dicritical if $A_\rho ^{(J_0)}(z,0)\ne 0$ and $\sigma _{\gamma _I}^{-1}(\gamma _I)$ a dicritical component of the total divisor $E'$.

Put $E_{J_0}:=\sigma _{\gamma _{I}}^{-1}(\gamma _I)\cap C_{J_0}$ and $F_{J_0}:=E_{J_0}\cap \{ \theta =0 \}$ and consider ${{\widetilde{\text {Sing}}}}(\hat{\eta }_{J_0}', F_{J_0})$ the adapted singular locus of $\hat{\eta }_{J_0}'$ relatively to $F_{J_0},$ it is a finite set, taking into account that the coefficients of $\hat{\eta }_{J_0}'$ belong to $\mathbb {R}[z][[\rho ]]$. Denote those elements contained in the regular part $\dot{F}_{J_0}:= F_{J_0}\cap \{ z>0 \}$ of $F_{J_0}$ as (in coordinates $(z^{(J_0)},\rho ^{(J_0)})$)

$$\begin{aligned} {{\widetilde{\text {Sing}}}}(\hat{\eta }_{J_0}', {F_{J_0}})\cap \dot{F}_{J_0}=\{ (\omega _i^{(J_0)},0): i=1,\ldots ,m_{J_0} \} \text {, with }0<\omega _i^{(J_0)}< \omega _j^{(J_0)}\text { if } i<j. \end{aligned}$$

The circles $\gamma _{I,i}:=\{ z=\omega _i^{(J_0)}, \rho =0 \}\subset E_{J_0}$ for each $i=1,\ldots , m_{J_0}$ are by definition the non-corner characteristic cycles in $C_{J_0}$. The circle $\gamma _{I,-\infty }:=\{ z=0, \rho =0 \}$ is by definition a corner characteristic cycle.

Gathering all the above objects, we define the family $\mathcal {D}':= \{ \gamma _I \}_{I\in \mathcal I'}$ of characteristic elements of $\mathcal {M}'$, where

$$\begin{aligned} \mathcal {I}'=\left( \mathcal {I} {\setminus }\{ {I} \} \right) \cup \left( \displaystyle \bigcup _{i=1}^{m_{J_0}} \{ {(I, i)} \} \right) \cup \{ {(I, -\infty ),(I, \infty )} \}. \end{aligned}$$

The elements of $\mathcal {D}'$ are subsets of $E'=(\pi ')^{-1}(0)$, once we identify $\gamma _L= \sigma _{\gamma _I}^{-1}(\gamma _L)$ for $L\in \mathcal {I} {\setminus } \{ I \}$. They are either the two points $\gamma _{\infty , \ldots , \infty }$ and $\gamma _{-\infty , \ldots , -\infty }$ (whose indices are denoted also by ${I^{\mathcal { M}'}_{ \infty }}$ and ${I^{\mathcal { M}'}_{ -\infty }}$, respectively) called the characteristic singularities of $\hat{\xi }$ in $\mathcal {M}'$ or circles (the characteristic cycles of $\hat{\xi }$ in $\mathcal { M}'$).

(ii) Second case. $\sigma _{\gamma _I}$ is centered at one of the characteristic cycles $\gamma _{I}\in \mathcal {D}$ with $I=(i_1,\ldots ,i_r)$. It can be a corner characteristic cycle (in which case $i_r=\pm \infty$) or not. The charts after the blowing-up $\sigma _{\gamma _I}$ are defined in a different manner in each case. In order to simplify the notation, name $I'=(i_1,\ldots , i_{r-1})$.

(a) When ${\gamma _I}$ is a corner characteristic cycle, it can be seen as $\{\rho ^{(J)}=0,z^{(J) }=0\}$ in a chart $C_{J}$ by (H3). Put $J_0=(I,0)$ and $J_\infty =(I,\infty )$. The set $\sigma _{\gamma _I} ^{-1}(\gamma _I)$ is covered by two new charts $(C_{J_\infty },(\theta ,z ^{(J_\infty )},\rho ^{(J_\infty )}))$ and $(C_{J_0},(\theta ,z^{(J_0)},\rho ^{(J_0)}))$, where the blowing-up $\sigma _{\gamma _I}$ is written as:

$$\begin{aligned}&\text {In }\quad C_{J_\infty }:&\left\{ \begin{matrix} \theta & =& \theta \\ z^{(J)}& =& z ^{(J_\infty )}\\ \rho ^{(J)}& =& \rho ^{(J_\infty )} z ^{(J_\infty )} \end{matrix} \right. , \quad \theta \in \mathbb R, \ z ^{(J_\infty )}, \rho ^{(J_\infty )} \ge 0, \end{aligned}$$

(17)

$$\begin{aligned}&\text {In }\quad C_{J_0}:&\left\{ \begin{matrix} \theta & =& \theta \\ z^{(J)}& =& \rho ^{(J_0)} z^{(J_0)}\\ \rho ^{(J)}& =& \rho ^{(J_0)} \end{matrix} \right. , \quad \theta \in \mathbb R, \ z^{(J_0)},\rho ^{(J_0)}, \ge 0. \end{aligned}$$

(18)

The new atlas is defined by $\mathcal {A}':=(\mathcal {A} {\setminus } \{ C_{J} \})\cup \{ C_{J_0},C_{J_\infty } \}$, where we have identified $M'{\setminus } \sigma _{\gamma _I} ^{-1}(\gamma _I)$ and $M{\setminus } \gamma _I$ via $\sigma _{\gamma _I}$.

To determine the new family $\mathcal {D}'$ of characteristic elements in this case, we write the transformed formal vector fields $\hat{\xi }^{(J_0)}:=(\sigma _{\gamma _{I}}|_{C_{J_0}})^{*}\hat{\xi }^{(J)}$, $\hat{\xi }^{(J_\infty )}:=(\sigma _{\gamma _{I}}|_{C_{J_\infty }})^{*}\hat{\xi }^{(J)}$ in the two charts. Both are similar and, in fact, to determine $\mathcal {D}'$ only one of the expressions is sufficient. Considering for instance the chart $C_{J_0}$, and with similar computations and notations as in the precedent paragraphs, we write (simplifying $(z,\rho )= (z^{(J_0)}, \rho ^{(J_0)})$)

$$\begin{aligned} \hat{\xi }^{(J_0)} = B_\theta ^{(J_0)}(z,\rho ) \frac{\partial }{\partial \theta }+ B_z ^{(J_0)}(z,\rho ) \frac{\partial }{\partial z}+ B ^{(J_0)}_\rho (z,\rho ) \frac{\partial }{\partial \rho }, \end{aligned}$$

(19)

where $B_\theta ^{(J_0)},B_z^{(J_0)},B_\rho ^{(J_0)}\in \mathbb {R}[z][[\rho ]]$ and $B_\theta ^{(J_0)}$ is a unit. The vector field associated to $\hat{\xi }^{(J_0)}$ is $\hat{\eta }_{J_0}:= (B_\theta ^{J_0})^{-1}B_{z}^{(J_0)} \frac{\partial }{\partial z}+ (B_\theta ^{J_0})^{-1}B_{\rho }^{(J_0)} \frac{\partial }{\partial \rho }$. We put

$$\begin{aligned} \hat{\eta }_{J_0}= \rho ^{n_1^{(J_0)}} z^{n_2^{(J_0)}}\hat{\eta }_{J_0}'= \rho ^{n_1^{(J_0)}} z^{n_2^{(J_0)}} \left( A_z^{(J_0)}(z,\rho )\frac{\partial }{\partial z}+ A_\rho ^{(J_0)}(z,\rho ) \frac{\partial }{\partial \rho }\right) , \end{aligned}$$

where the natural numbers $n_k^{(J_0)}$ for $k=1,2$ are defined as in case (i). The vector field $\hat{\eta }_ {J_0}'$ is the reduced associated vector field. We distinguish the cases when $\sigma _{\gamma _{I}}$, or the component $E_{J_0}:=\sigma _{\gamma _{I}}^{-1}(\gamma _I)$, is dicritical $(A_\rho ^{(J_0)}(z,0)\ne 0 )$ or non-dicritical $(A_\rho ^{(J_0)}(z,0)\equiv 0 )$. Put $F_{J_0}:= E_{J_0}\cap \{ \theta =0 \}, \ \dot{F}_{J_0}:=F_{J_0}\cap \{ z>0 \}$ and denote

$$\begin{aligned} {{\widetilde{\text {Sing}}}}(\hat{\eta }_{J_0}', { F_{J_0}})\cap \dot{F}_{J_0}=\{ (\omega _i^{(J_0)},0): i=1,\ldots ,m_{J_0} \} \text {, with }0<\omega _i^{(J_0)}< \omega _j^{(J_0)}\text { if } i<j. \end{aligned}$$

With these data, we set:

$$\begin{aligned} \gamma _{I,i}:= & \mathbb {S}^1 \times \{ (z^{(J_0)},\rho ^{(J_0)})=(\omega _i ^{(J_0)},0) \}, i =1, \ldots m_{J_0}\\ \gamma _{I,-\infty }:= & \mathbb {S}^1 \times \{ (z^{(J_0)},\rho ^{(J_0)})=(0,0) \} \subset C_{J_0}\\ \gamma _{I,\infty }:= & \mathbb {S}^1 \times \{ (z^{(J_\infty )},\rho ^{(J_\infty )})=(0,0) \} \subset C_{J_\infty } \end{aligned}$$

and we define the family of characteristic elements of $\mathcal { M}'$ as $\mathcal {D}':= \{ \gamma _I \}_{I\in \mathcal I}$, where

$$\begin{aligned} \mathcal {I}'=\left( \mathcal {I}{\setminus } \{ I \} \right) \cup \displaystyle \{ (I,i) \}_{i=1}^{m_{J_0}} \cup \{ (I,-\infty ) \} \cup \{ (I,\infty ) \}, \end{aligned}$$

again identifying $\gamma _L$ with $\sigma _{\gamma _{I}}^{-1}(\gamma _L)$ for $L\in \mathcal I {\setminus } \{ I \}$. Notice that, among the new characteristic cycles, $\gamma _{I,\infty }$, $\gamma _{I,-\infty }$ are corner cycles and the other ones are non-corner characteristic cycles.

(b) When $\gamma _I$ is a non-corner characteristic cycle (that is, by (H2), when $I=(i_1, \ldots , i_{r})$ with $i_{r}\ne \pm \infty$), it can be seen as the set $\gamma _I=\{z^{(J )}=\omega ^{(J)}_{k}, \rho ^{(J)}=0\}$ for some $\omega _k^{(J )}$ in the domain of $z^{(J)}$ of a chart $C_{J}$, by (H3). Set $J_{-\infty }:=(I,-\infty )$, $J_\infty :=(I,\infty )$ and $J_0:=(I,0)$. The blowing-up $\sigma _{\gamma _I}:(M',E')\longrightarrow (M,\gamma _I)$ of $\gamma _I$ is given in three new charts $(C_u,(\theta , z^{(u)},\rho ^{(u)}))$, for $u\in \{ J_\infty , J_0, J_{-\infty } \}$, by

$$\begin{aligned}&\text {In } C_{J_\infty }:&\left\{ \begin{matrix} \theta & =& \theta \\ z^{(J)}& =& z^{(J_ \infty )}- \omega ^{(J)}_{k}\\ \rho ^{(J)}& =& \rho ^{(J_\infty )} z ^{(J_\infty )} \end{matrix} \right. \qquad \ \theta \in \mathbb R, \rho ^{(J_ \infty )}, z ^{(J_\infty )} \ge 0. \end{aligned}$$

(20)

$$\begin{aligned}&\text {In } C_{J_ 0}:&\left\{ \begin{matrix} \theta & =& \theta \\ z^{(J)}& =& \rho ^{(J_ 0)} (z^{(J_0)} - \omega ^{(J)}_{k} )\\ \rho ^{(J)}& =& \rho ^{(J_0)} \end{matrix} \right. \quad \theta \in \mathbb R,z^{(J_0)} \in \mathbb {R}, \rho ^{(J_0)} \ge 0. \end{aligned}$$

(21)

$$\begin{aligned}&\text {In } C_{J_{-\infty }}:&\left\{ \begin{matrix} \theta & =& \theta \\ z^{(J)}& =& -z^{(J_{ -\infty })}+ \omega ^{(J)}_{k} \\ \rho ^{(J)}& =& \rho ^{(J_{-\infty })} z^{(J_{ -\infty })} \end{matrix} \right. \quad \theta \in \mathbb R, \rho ^{(J_{ -\infty })}, z^{(J_{ -\infty })} \ge 0. \end{aligned}$$

(22)

The new atlas is $\mathcal {A}':= (\mathcal {A}{\setminus } \{ C_J \}) \cup \{ C_{J_0}, C_{J_\infty }, C_{J_{-\infty }} \}$. The family $\mathcal {D}'$ of characteristic elements of $\hat{\xi }$ in $\mathcal M'$ is defined analogously as in case (a), studying the corresponding transformed vector field $\hat{\xi }^{(J_0)}= (\sigma _{\gamma _I}|_{C_{J_0}})^*\hat{\xi }^{(J)}$, its associated two-dimensional vector field $\hat{\eta }_{J_0}$ and the adapted singular locus of the reduced associated vector field $\hat{\eta }_ {J_0}'$ relatively to $F_{J_0}=C_{J_0}\cap \sigma _{\gamma _I}^{-1}(\gamma _I)\cap \{ \theta =0 \}$. We just observe the following:

• The charts $C_{J_\infty }$ and $C_{J_{-\infty }}$ are corner charts and the curves $\gamma _{J_\infty }=\{z ^{(J_\infty )}=0,\rho ^{(J_\infty )}=0 \}$ and $\gamma _{J_{-\infty }}=\{z^{(J_{-\infty })}=0, \rho ^{(J_{-\infty })}=0\}$ are corner characteristic cycles.

• The chart $C_{J_0}$ is a non-corner chart and contains the new non-corner characteristic cycles $\gamma _{I,i}$ where $\gamma _{I,i}=\mathbb {S}^1\times \{ (\omega _i^{(J_0)},0) \}, \ i=1, \ldots , m_{J_0}$ being ${{\widetilde{\text {Sing}}}}(\hat{\eta }_{J_0}',F_{J_0})=\{ (\omega _i^{(J_0)},0),\ i=1,\ldots , m_{J_0} \}$.

From the construction, we can check that the hypothesis (H1-H3) are fulfilled for $\mathcal M'$. Thus, we have defined admissible sequences of blowing-ups of $\hat{\xi }$ of any length.

Remark 3.6

By construction, a non-corner characteristic cycle $\gamma _I\in \mathcal {D}$ is defined in some chart $C_J$ by equations $\gamma _I=\{ z^{( J)}=\omega ^{(J)}_k, \rho ^{( J)}=0 \}$ for some $k\in \mathbb {N}_{\ge 1}$. It may happen that the same characteristic cycle $\gamma _I$ is defined in a corner chart $C_{{\tilde{J}}}$ by $\{ z^{({\tilde{J}})}=0, \rho ^{({\tilde{J}})}=c \}$, for some $c\in \mathbb R$.

Remark 3.7

Notice that for any given sequence of admissible blowing-ups $\mathcal { M}=(M,\pi , \mathcal {A},\mathcal {D})$ and for any chart $C_J$ of $\mathcal {A}$, the associated vector field $\hat{\eta }_{ J}$ is not identically zero. This can be seen from the construction of $\mathcal { M}$ and using Remark 3.1.

3.4 Adapted Reduction of Singularities

Recall (see the book [5]) that a formal vector field $\chi =A(x,y)\frac{\partial }{\partial x}+B(x,y)\frac{\partial }{\partial y}$ at $(\mathbb {R}^2,0)$ has a (real) simple singularity if $\text {Sing}(\chi )=\{0\}$, the eigenvalues $\lambda _1,\lambda _2$ of the linear part $D\chi (0)$ are real and at least one of them is different from zero, for instance $\lambda _2\ne 0$, and $\frac{\lambda _1}{\lambda _2}\notin \mathbb {Q}_{>0}$. In this case, $\chi$ has exactly two formal invariant curves, also called separatrices, which are tangent to the corresponding eigenspaces, non-singular and mutually transverse. We need an extended notion of simple singularity, also taken from that reference, that takes into account the existence of a divisor and the possibility that the singularity is not isolated.

Definition 3.8

Let $F= \{ xy^\epsilon =0 \}$, where $\epsilon \in \{ 0,1\}$, be a normal crossing divisor at $0\in \mathbb {R}^2$. A formal vector field $\chi$ at $(\mathbb {R}^2,0)$ has an adapted simple singularity relatively to F if one of the two following situations occurs:

• (1) $\text {Sing}(\chi )=\{0\}$, the singularity is simple and each component of F is invariant for $\chi$ (thus, if $\epsilon =1$, the two components of F are the two separatrices).

• (2) $\epsilon =0$, there is a formal non-singular curve $\Gamma$ transversal to $F=\{x=0\}$ given by an equation $\Gamma =\{ y-\hat{g}(x)=0 \}$ contained in $\text {Sing}(\chi )$, and $\chi =( y-\hat{g}(x))^r\bar{\chi }$, with $r\ge 1$, such that either $\bar{\chi }$ is non-singular at 0 and F is the only invariant curve of $\bar{\chi }$ through 0, or $\bar{\chi }$ has a simple singularity at 0 and the set of separatrices of $\bar{\chi }$ at 0 is $\{ F,\Gamma \}$.

To distinguish the two cases of this definition, in the situation of (2), we say that $\chi$ has a non-saturated adapted simple singularity. Usually in this situation, one divides $\chi$ by an equation of $\text {Sing}(\chi )$ to get the situation in (1) or a non-singular point. However for us, the vector field $\chi$ will come from some three dimensional vector field, hence it will be important to keep unaltered the singular locus placed outside the divisor.

Before introducing the reduction of singularities of $\hat{\xi }$ adapted to our problem, we recall Seidenberg’s Theorem ([29]) of reduction of singularities of a two dimensional analytic (or formal) vector field $\xi$, following the lines of the book [5]. In this reference, it is assumed that the vector field is saturated, i.e. that $\chi$ has an isolated singularity at the origin. For us, it is important to consider the non-saturated case: that $\chi$ writes as $\chi =f\bar{\chi }$, where f is non-zero, non-unit and a generator of $\text {Sing}(\chi )$. Moreover, we cannot “saturate" $\chi$ just by dividing by f since we treat the formal case and we want to preserve the analytic nature of the given coordinates. Instead, we adapt the result in [5] to the non-saturated case, which only involves a slightly modification and encompasses both a reduction of singularities of the singular locus of $\chi$ and a reduction of singularities of $\bar{\chi }$. For the sake of completeness, we provide here a precise statement and we sketch the modifications to be made for its proof.

Theorem 3.9

Let $\chi$ be a formal vector field at $(\mathbb {R}^2,0)$ not identically zero, saturated or not, $F^{(0)}$ be a normal crossings divisor and $0\in {{\widetilde{\text {Sing}}}}(\chi ,F^{(0)})$. Then there is a composition of a finite number of punctual blowing-ups $\pi :({\tilde{N}},{\tilde{F}})\rightarrow (\mathbb {R}^2,F^{(0)})$ fulfilling the following conditions:

(a):

For any point $q\in {\tilde{F}}=\pi ^{-1}(F^{(0)})$, if $\chi _q'$ is the strict transform of $\chi$ by $\pi$ at q (that is, $\chi _q'=\frac{1}{u^kv^l} \pi ^*(\chi )$, where $uv^\epsilon$ is a local reduced equation of ${\tilde{F}}$ at q ($\epsilon =0$ or 1) and k, l are maximal so that $\chi _q'$ has no pole), then $q\in {{\widetilde{\text {Sing}}}}(\chi _q', {\tilde{F}})$ if and only if $q\in \text {Sing}(\chi _q')$.

(b):

If $q\in {\tilde{F}}$ is a singular point of $\chi _q'$, then q is an adapted simple singularity relatively to ${\tilde{F}}$ (cf. Definition 3.8).

(c):

Any dicritical component is isolated as a dicritical component (i.e. any other component that intersects it is non dicritical).

Proof

In the case where $\chi$ has as an isolated singularity at 0, the existence of the reduction of singularities $\tau$ is given by the result in [5], where one eliminates points in the adapted singular locus relatively to the divisor that are not singular points (to get (a)). In the case where the singular locus $S:=\text {Sing}(\chi )$ of $\chi$ at 0 is not reduced to $\{ 0\}$ (thus S is a finite union of formal curves), we first consider a reduction of singularities $\psi :( N^{(1)}, F^{(1)})\rightarrow (\mathbb {R}^2, F^{(0)})$ of S. Then, let $\chi _1'$ be the strict transform of $\chi$ by $\psi$, we blow up any $q\in {{\widetilde{\text {Sing}}}}(\chi _1',F^{(1)})$ that is not a singular point to get (a). After that, we may assume that such strict transform, named $\chi _q'$, either has an isolated singularity at q (and hence we apply again [5]) or q is a point in the strict transform $S_{q}^{(1)}$ of the curve S by $\psi$. In this last case, there are coordinates (x, y) at q such that $F^{(1)}=\{x=0\}$, and $\chi _q'$ is written as $\chi _q'=(y-{\hat{g}}(x))^r \bar{\chi }_q'$, where $\{ y-{\hat{g}}(x)=0 \}$ is an equation of $S_{q}^{(1)}$, $r\ge 1$ and $\bar{\chi }_q'$ has at most an isolated singularity at q.

If $\bar{\chi }_q'(q)=0$, after a reduction of singularities of $\bar{\chi }_q'$, we may assume that q is a simple singularity of $\bar{\chi }_q'$. By further blowing-ups, we separate ${\tilde{S}}_{q}^{(1)}$ from the two separatrices of $\bar{\chi }_q'$ unless one of them coincides with ${\tilde{S}}_{q}^{(1)}$. We will get in this way adapted simple singularities of (the transform of) $\tilde{\chi }_q'$ either saturated (cf. Definition 3.8-(1)) or non saturated (cf. Definition 3.8-(2)).

When $\bar{\chi }_q'(q) \ne 0$, if $\Gamma$ is the formal solution of $\bar{\chi }'_q$ through q, a new blowing-up at q produces an adapted simple singularity for the transform of $\bar{\chi }_q'$ at the point corresponding to the tangent line of $\Gamma$. If $\Gamma$ coincides with $\tilde{S}_{q}^{(1)}$, we get an adapted simple singular point for ${\tilde{\chi }}_q'$. Otherwise, by further blowing-ups, we separate $\Gamma$ from $\tilde{S}_{q}^{(1)}$ and we get either adapted simple singularities or points in the situation already treated.

Note that condition (c) is obtained as a consequence of the result in [5] since only normal crossings are allowed. $\square$

Now, we can state the result which gives the reduction of singularities of a formal normal form $\hat{\xi }$ of a vector field $\xi \in \mathcal {H}^3$ with isolated singularity.

Proposition 3.10

(Adapted resolution of singularities) Let $\hat{\xi }$ be a formal vector field written as in Eq. (2) with isolated singularity at $0\in \mathbb {R}^3$. Then there exists a sequence of admissible blowing-ups $\mathcal {M}=(M,\pi ,\mathcal {A}, \mathcal {D} )$ for $\hat{\xi }$ with $\mathcal {A}=\{ C_J \}_{J\in \mathcal J}$, $\mathcal {D}=\{ \gamma _I \}_{I\in \mathcal I}$ and total divisor $E=\pi ^{-1}(0)$ such that

• (1) For $J\in \{ I_\infty ^\mathcal {M}, I_{-\infty }^\mathcal {M} \}$, the transformed vector field $\hat{\xi }^{(J)}=(\pi |_{C_J})^*\hat{\xi }$ satisfies $\hat{\xi } ^{(J)}(z^{(J)}) =(z^{(J)})^t \cdot G$ where $t\ge 1$ and G is a unit in $\mathbb {R}[[x^{(J)},y^{(J)},z^{(J)}]]$.

• (2) For any $J\in \mathcal J {\setminus } \{ I_\infty ^\mathcal {M}, I_{-\infty }^\mathcal {M} \}$, the singularities of the reduced associated vector field $\hat{\eta }_J'$ are adapted simple singularities relatively to the divisor $E\cap C_J\cap \{ \theta =0 \}$.

• (3) If $E_0$ is a dicritical component of E, then $E_0$ is isolated as a dicritical component (i.e. any other component that intersects $E_0$ is non dicritical). Moreover, for any $J \in \mathcal {J}{\setminus } \{ I_\infty ^\mathcal {M}, I_{-\infty }^\mathcal {M} \}$, one has ${{\widetilde{\text {Sing}}}}(\hat{\eta }_J', F_{0,J})=\emptyset$ where $F_{0,J}=E_0\cap C_J\cap \{ \theta =0 \}$, in particular, $\hat{\eta }_J'$ is everywhere transversal to $F_{0,J}$.

Proof

From Remark 3.1, there exists a term $c_jz^j$ in the coefficient $\hat{\xi } (z)$ with $c_j\ne 0$. Assume, without loss of generality, that j is the minimum exponent with this condition. Notice that $j>0$. Write $\hat{\xi } (z)\in \mathbb {R}[[x,y,z]]$ as

$$\begin{aligned} \hat{\xi } (z)= Z(x^2+y^2,z)=z^{t_0} G(x,y,z)=z^{t_0} \displaystyle \sum _{k=\nu (G)}^{\infty } G_k(x,y,z), \end{aligned}$$

where $G_k$ is an homogeneous polynomial of degree k for each k, $t_0\ge 0$ is defined as the maximum integer such that $z^{t_0}$ divides $\hat{\xi } (z)$ and $\nu (G)$ is the order of G as a series as defined in the Introduction. Then $G_{j-{t_0} }(x,y,z)$ contains the monomial $c_j z ^{j-t_0}$ (notice that $j\ge t_0$ and the equality holds if and only if $\nu (G)=0$). Consider the first blowing-up $\sigma _0$ and study $\hat{\xi }^{(\infty )}(z^{(\infty )})$, where $\hat{\xi }^{(\infty )}=(\sigma _{0}|_{C_\infty })^*\hat{\xi }$. Omitting super-indices for the coordinates $(x^{(\infty )}, y^{(\infty )}, z^{(\infty )})$, we have:

$$\begin{aligned} \hat{\xi }^{(\infty )} (z)= z^{t_0}\displaystyle \sum _{k=\nu (G)}^{\infty } G_k(x,y,1)z^k= z^{t_1} \displaystyle \sum _{k=\nu (G)}^{\infty } G_k(x,y,1)z^{k-\nu (G)}, \end{aligned}$$

where $t_1=t_0+\nu (G)\ge t_0$. Rewrite the series $G^{(1)}:=\sum _{k=\nu (G)}^{\infty } G_k(x,y,1)z^{k-\nu (G)}$ in homogeneous components:

$$\begin{aligned} \hat{\xi }^{(\infty )} (z)=z^{t_1}G^{(1)}(x,y,z) =z^{t_1} \displaystyle \sum _{k=\nu (G^{(1)})}^{\infty } G_k^{(1)}(x,y,z). \end{aligned}$$

If $j=t_1$, we see that $G^{(1)}_0=c_j$ and thus $G^{(1)}$ is a unit, which gives statement (1) of the proposition for $t=t_1$. Otherwise, if $t_1<j$, we see that $G^{(1)}_{j-t_0}(x,y,z)$ contains the term $c_jz^{j-t_1}$. Notice that, in this case, we have $t_1\ge t_0$ since, otherwise, if $t_1=t_0$ then $\nu (G)=0$ and $j=t_0=t_1$. Thus, $j-t_0>j-t_1\ge 0$. By recurrence over $j-t_0$, there exists an admissible sequence of blowing-ups $\widetilde{\mathcal {M}}= ({{\widetilde{M}}}, {{\widetilde{\pi }}}, {{\widetilde{\mathcal {A}}}}, {{\widetilde{\mathcal {D}}}})$ with $\widetilde{\pi }$ a composition of s blowing-ups at the corresponding characteristic singularities $\gamma _{I_\infty ^{\mathcal { M}_i}}$ such that, defining $t_0,t_1, \ldots , t_s$ as above, we have $j=t_s$. We conclude (1) for ${{\widetilde{\pi }}}^*\hat{\xi }$ at the characteristic singularity $\gamma _{I_\infty ^{\widetilde{\mathcal {M}}}}$ with $t=t_s$. Analogously, up to blowing-up repeatedly the characteristic singularity $\gamma _{I_{-\infty }^{\widetilde{\mathcal {M}}}}$, we may assume that (1) holds at $\gamma _{I_{-\infty }^{\widetilde{\mathcal {M}}}}$.

According to the construction of sequences of admissible blowing-ups in the Sect. 3.3, ${{\widetilde{\mathcal {A}}}}$ is composed by the two charts $C_{I_{-\infty }^{\widetilde{\mathcal {M}}}}$ and $C_{I_{\infty }^{\widetilde{\mathcal {M}}}}$ and a finite number of charts named:

$$\begin{aligned} \{ C_J \}_{J\in {\widetilde{\mathcal {J}}}_{0}}, \ \text {where}\ {\widetilde{\mathcal {J}}}_{0}=\{ 0,(\infty ,0),\ldots , (\infty ,\ldots ,\infty ,0),(-\infty ,0),\ldots ,(-\infty ,\ldots , -\infty ,0) \} \end{aligned}$$

with coordinates of the form $(\theta ,z,\rho )\in \mathbb {R}\times (\mathbb {R}_{\ge 0})^2$, except for the first one with z taking values in $\mathbb {R}$. For any $J\in {{\widetilde{\mathcal {J}}}}_{0}$, consider the transformed vector field $\hat{\xi }^{(J)}=({{\widetilde{\pi }}}|_{C_J})^*\hat{\xi }$ and the corresponding reduced associated vector field $\hat{\eta }_J'$. Denote by ${\tilde{E}}:= {\tilde{\pi }}^{-1} (0)$ the total divisor of ${{\widetilde{\mathcal {M}}}}$. Notice that the coordinate $\theta$ is well defined in the union $U=\bigcup _{J\in {{\widetilde{\mathcal { J}}}}_{0}} C_{J}$ so that ${\tilde{F}}:={\tilde{E}}\cap \{ \theta =0 \}\cap U$ has a perfect sense. In fact, ${\tilde{F}} ={\tilde{E}} \cap \overline{\pi ^{-1}(\{y=0, x>0\})}$. Now, given $J\in {{\widetilde{\mathcal {J}}}}_{0}$, we apply Theorem 3.9 at each point $a\in {{\widetilde{\text {Sing}}}}(\hat{\eta }_J',\tilde{F})$ in order to obtain a reduction of singularities $\tau _a$ of the two dimensional vector field $\hat{\eta }_{J}'$ at a adapted to ${\tilde{F}}$.

Notice that in the sequence of blowing-ups that Theorem 3.9 provides, we start blowing up with center at points $a\in {{\widetilde{\text {Sing}}}}(\hat{\eta }_J', {\tilde{F}})$ for the different $J\in {\widetilde{\mathcal {J}}}_{0}$. The points in ${{\widetilde{\text {Sing}}}}(\hat{\eta }_J', {\tilde{F}})$ correspond exactly to the family of characteristic cycles of ${{\widetilde{\mathcal {M}}}}$ (elements of ${{\widetilde{\mathcal {D}}}}$). Considering admissible blowing-ups $\sigma _{\gamma _{I}}$ centered at those $\gamma _I\in {{\widetilde{\mathcal {D}}}}$, the restriction $\sigma _{\gamma _{I}}|_{\{\theta =0\}}$ is exactly the blowing-up centered at the corresponding point $\gamma _I\cap \{ \theta =0 \}$ of the two-dimensional vector field $\hat{\eta }_J'$. Moreover, this property repeats for the subsequent points to be blown up to achieve $\tau _a$ and the corresponding strict transform of $\hat{\eta }_J'$. In other words, having defined the sequence of blowing-ups $\tau _a$ as above, satisfying (a), (b) and (c) for any $a\in {{\widetilde{\text {Sing}}}}(\hat{\eta }_J',{\tilde{F}})$ and for any $J\in {{\widetilde{J}}}_0$, the composition of these sequences of two dimensional blowing-ups $\tau _a$ provides a sequence of admissible blowing-ups $\mathcal { M}=(M,\pi , \mathcal {A}, \mathcal {D})$ factorizing through ${{\widetilde{\pi }}}$ (i.e. $\pi ={{\widetilde{\pi }}}\circ \pi '$) such that $\mathcal { M}$ satisfies (2) and (3) of the statement. Since $\pi '$ does not modify the characteristic singularities $\gamma _{I_{-\infty }^{\widetilde{\mathcal {M}}}}, \gamma _{I_{-\infty }^{\widetilde{\mathcal {M}}}}$, we have also (1), and we are done. $\square$

Remark 3.11

Notice that after an adapted reduction of singularities, the non-corner characteristic cycles that we obtain are contained in non-dicritical components of the total divisor.

3.5 Behavior of Jet Approximations of Normal Forms Under Blowing-Ups

In this section, we study the effect of sequences of admissible blowing-ups to the jet approximations $\xi _{\ell }$ of the formal normal form $\hat{\xi }$, for convenient values of $\ell$. First, we establish the jet dependence of the transform of $\hat{\xi }$ by such blowing-ups in the different charts.

Proposition 3.12

Let $\hat{\xi }$ be a formal normal form of $\xi \in \mathcal H_3$. Consider an admissible sequence of blowing-ups $\mathcal {M}=(M,\pi ,\mathcal {A}, \mathcal {D})$ for $\hat{\xi }$ of length $l>0$, with $\mathcal {A}=\{ C_J \}_{J\in \mathcal J}$. For every $J\in \mathcal J$ and for every $k\ge 1$, if u is a coordinate of the chart $C_J$ such that $\{ u=0 \}\subset E=\pi ^{-1}(0)$, then we have

$$\begin{aligned} j_{k}^u(\hat{\xi }^{(J)})=j_k^u((\pi |_{C_J})^{*} j_{k+l+1} (\hat{\xi })). \end{aligned}$$

(23)

Proof

The proof uses the following standard fact.

Fact

Let $\eta$ be a vector field with coefficients in $A[[x_1,\ldots , x_n]]$ and let $\tau$ be a quadratic morphism of the form $\tau (x_1,\ldots , x_n)=(x_1x_i,\ldots , x_{i-1}x_i, x_i, x_{i+1}x_i, \ldots , x_nx_i)$. Then,

$$\begin{aligned} \begin{aligned} j_k^{x_i}(\tau ^*\eta )&=j_k ^{x_i} (\tau ^*j_{k+1}(\eta )) \\ j_k^{x_j}(\tau ^*\eta )&= j_k ^{x_j} (\tau ^*j_{k}^{x_j}(\eta ))=j_k ^{x_j} (\tau ^*j_{k+1}^{x_j}(\eta )), \ j\ne i. \end{aligned} \end{aligned}$$

(24)

We proceed by induction on the length l of $\mathcal { M}$. If $l=0$, that is, $\pi = \sigma _{0}$ is the blowing-up of the origin $0\in \mathbb {R}^3$ described in Sect. 3.2. We have (with simplified notation $\rho :=\rho ^{(0)}, z:=z^{(0)}$)

$$\begin{aligned} \begin{aligned} j^\rho _{k}((\sigma _0|_{C_0})^*\hat{\xi })&=j^\rho _{k}( (\sigma _0|_{C_0})^*j_{k+1} (\hat{\xi }) ),\\ j^z_k((\sigma _0|_{C_\infty })^*\hat{\xi })&= j^z_{k}( (\sigma _0|_{C_\infty })^*j_{k+1} (\hat{\xi }) ),\\ j^z_k((\sigma _0|_{C_{-\infty }})^*\hat{\xi })&= j^z_{k}( (\sigma _0|_{C_{-\infty }})^*j_{k+1} (\hat{\xi }) ), \end{aligned} \end{aligned}$$

(25)

which proves the result.

Suppose $l>0$ and that $\pi ={\widetilde{\pi }} \circ \sigma _{\gamma _I}$, where $\sigma _{\gamma _I}$ is the blowing-up centered at some characteristic element $\gamma _I$ of a sequence of admissible blowing-ups ${\widetilde{\mathcal {M}}}=({\widetilde{M}},{\widetilde{\pi }},\widetilde{ \mathcal {A}}, \widetilde{\mathcal {D}})$ of length $l-1$. It is enough to study the transform $\hat{\xi } ^{(J)}$ in the charts $C_J$ when $\sigma _{\gamma _I} ^{-1}(\gamma _I)\cap C_J\ne \emptyset$, since the map $\sigma _{\gamma _I}$ is an isomorphism out of $\sigma _{\gamma _I} ^{-1}(\gamma _I)$. According to the construction of $\mathcal {M}$ from $\widetilde{\mathcal {M}}$ and using the same notations as in Sect. 3.3, we have several cases:

• (1) The point $\gamma _I$ is the origin of a chart $(C_{J_I},(x^{(J_I)}, y^{(J_I)}, z^{(J_I)} ))$ of $\widetilde{\mathcal {A}}$ (for instance $I=I ^{\widetilde{\mathcal {M}} }_{ \infty }$) where $z^{(J_I)}=0$ is the equation of the divisor $\widetilde{E}\cap C_{J_I}$, and $J=I ^{{\mathcal { M}} }_{ \infty }=(\infty ,{\mathop {\ldots }\limits ^{s}},\infty )$. In this case, $u=z^{(J)}$ is the only coordinate of the chart $C_J$ in the conditions of the statement. Using the induction hypothesis $j_{k'}^{z^{(J_I)}}(\hat{\xi } ^{(J_I)})=j_{k'}^{z^{(J_I)}} ( (\pi |_{C_{J_I}})^*j_{k'+(l-1)+1}(\hat{\xi }) )$ for $k'=k+1$, we have that

  $$\begin{aligned} j_k^{u}(\hat{\xi }^{(J)})&=j^u_k ((\sigma _{\gamma _{I}}|_{C_J})^* \hat{\xi }^{(I)}) = j^u_k ((\sigma _{\gamma _{I}}|_{C_J})^* j_{k+1}^{z^{(J_I)}}(\hat{\xi }^{(I)})) \\&= j^u_k ((\sigma _{\gamma _{I}}|_{C_J})^* j_{k+1}^{z^{(J_I)}}(({\tilde{\pi }}|_{C_I})^* (j_{(k +1) + (l-1)+1} (\hat{\xi }))) \\&= j^u_k ((\sigma _{\gamma _{I}}|_{C_J})^* (({\tilde{\pi }}|_{C_I})^* (j_{k + l+1} (\hat{\xi }))) \\&= j^u_k (( \pi |_{C_J})^* (j_{k + l+1} (\hat{\xi })) \end{aligned}$$

• (2) The point $\gamma _I$ is the origin of a chart $(C_{J_I},(x^{(J_I)}, y^{(J_I)}, z^{(J_I)} ))$ of $\widetilde{\mathcal { A}}$ where $z^{(J_I)}=0$ is the equation of the divisor $\widetilde{E}\cap C_{J_I}$ and $\sigma _{\gamma _{I}}|_{C_J}: C_J\rightarrow C_{J_I}$ has the same expression as (4) for $\sigma _0$, considering coordinates $(\theta ,z^{(J)},\rho ^{(J)})$ for $C_J$ and with the obvious change of notation. Notice that in $C_J$ the two coordinates $u=\rho ^{(J)}$ and $u=z^{(J)}$ are in the conditions of the statement. By the induction hypothesis, renaming $z=z^{(J_I)}$ for simplicity, we have, for any $k\ge 1$, that $j^{z}_{k}(\hat{\xi } ^{(J_I)})= j^{z}_{k} (({\tilde{\pi }}|_{C_{J_I}})^*j_{k+l} (\hat{\xi } ) )$. By the fact that $j_k(\chi )=j_k(j_k^z(\chi ))$ for any vector field $\chi$, we also have $j_{k}(\hat{\xi } ^{(J_I)})= j_{k} (({\tilde{\pi }}|_{C_{J_I}})^*j_{k+l} (\hat{\xi } ) )$. From this last equality, the result follows for $u=\rho ^{(J)}$ similarly to the case of the first blowing-up $\sigma _0$. For $u=z^{(J)}$, it is a consequence of the second equation of (24).

• (3) ${\gamma _I}$ is a characteristic cycle of $\widetilde{ \mathcal {M}}$. Taking into account Remark 3.6, we may assume $\gamma _I\subset \{ \rho ^{( J_I)}=0 \}$ for some chart $(C_{J_I},(\theta , z^{(J_I)},\rho ^{(J_I)}))\in \tilde{\mathcal {A}}$. Let us put for simplicity $(z,\rho )=(z^{(J_I)},\rho ^{(J_I)})$. We distinguish two cases:

  • (a) $\gamma _I$ is a corner characteristic cycle. In this case, $\sigma _{\gamma _I}^{-1}(\gamma _I)$ is covered by two charts $(C_{J}, (\theta ,z^{(J)}, \rho ^{(J)}))$ with $J=J_\infty ,J_0$, for which the expression of $\sigma _{\gamma _I}$ is given by (17) and (18), respectively. By symmetry, both are treated similarly, and we assume the case $J=J_\infty$. Notice that the coordinates $u=z^{(J)}$ and $u=\rho ^{(J)}$ are in the condition of the statement. For $u=z^{(J)}$, we have, for any $k\ge 1$:

    $$\begin{aligned} \begin{aligned} j_k^u(\hat{\xi }^{(J)})&= j_k^u((\sigma _{\gamma _I}|_{C_J})^{*} j_{k+1}(\hat{\xi }^{(J_I)}))= j_k^u((\sigma _{\gamma _I}|_{C_J})^{*} j_{k+1}(j_{k+1}^z(\hat{\xi }^{(J_I)}))) \\&= j_k^u( (\sigma _{\gamma _I}|_{C_J})^{*} j_{k+1} ( j_{k+1}^z ( ({\widetilde{\pi }}|_{C_{J_I}})^*(j_{k+l+1}(\hat{\xi })) ) ) ) \\ &=j_k^u( (\sigma _{\gamma _I}|_{C_J})^{*} j_{k+1} (({\widetilde{\pi }}|_{C_{J_I}})^*(j_{k+l+1}(\hat{\xi })) ) )\\&= j_k^u( (\sigma _{\gamma _I}|_{C_J})^{*} ({\widetilde{\pi }}|_{C_{J_I}})^*(j_{k+l+1}(\hat{\xi })) ) = j_k^u((\pi |_{C_J})^*(j_{k+l+1}(\hat{\xi }))). \end{aligned} \end{aligned}$$

    (26)

    Here, we have used the first formula of Eq. (24) for the quadratic map $\sigma _{\gamma _{I}}$ in the first and fifth equalities, general properties of jets (cf. Sect. 1) in the second and fourth equalities and the induction hypothesis in the third equality. This proves (23) for $u=z^{(J)}$. For $u=\rho ^{(J)}$, we have, for any $k\ge 1$:

    $$\begin{aligned} \begin{aligned} j_k^u(\hat{\xi }^{(J)})&= j_k^u((\sigma _{\gamma _I}|_{C_J})^{*} j_{k+1}^\rho (\hat{\xi }^{(J_I)})) \\ &= j_k^u( (\sigma _{\gamma _I}|_{C_J})^{*} j_{k+1}^\rho ( ({\widetilde{\pi }}|_{C_{J_I}})^*(j_{k+l+1}(\hat{\xi })) ) ) \\&= j_k^u( (\sigma _{\gamma _I}|_{C_J})^{*} ({\widetilde{\pi }}|_{C_{J_I}})^*(j_{k+l+1}(\hat{\xi })) )= j_k^u((\pi |_{C_J})^*(j_{k+l+1}(\hat{\xi }))), \end{aligned} \end{aligned}$$

    (27)

    where we have used the second formula of (24) for the quadratic map $\sigma _{\gamma _{I}}$ in the first and third equality and the induction hypothesis in the second equality. This proves (23) for $u=\rho ^{(J)}$.

  • (b) $\gamma _I$ is a non-corner characteristic cycle. In this case $\sigma _{\gamma _{I}}^{-1}(\gamma _I)$ is covered by three charts $C_{J_{\infty }}, \ C_{J_0}$ and $C_{J_{-\infty }}$, for which the expression of $\sigma _{\gamma _{I}}$ is given by Eqs. (20), (21) and (22), respectively. In the chart $(C_{J_\infty },(\theta , z ^{(J_\infty )}, \rho ^{(J_\infty )}))$, the two coordinates $u=z^{(J_\infty )}$ and $u=\rho ^{(J_\infty )}$ are in the hypothesis of the statement. The proof of the result is analogous to the one in case (a), namely Eqs. (26) and (27). The chart $C_{J_{-\infty }}$ is similar to $C_{J_\infty }$. Finally, in the chart $(C_{J_0},(\theta ,z^{(J_0)},\rho ^{(J_0)}))$ only $u=\rho ^{(J_0)}$ is in the hypothesis of the statement. The proof for this coordinate is just the same sequence of equalities as in (26) with the interchange of the role of the coordinates z and $\rho$ in $C_{J_I}$.

$\square$

Now, let us discuss the validity of Proposition 3.12 for the jets approximations of the normal form $\xi _{\ell }$.

Consider the first blowing-up $\sigma _{0}$ at $0\in \mathbb {R}^3$, a singular point of $\xi _{\ell }$ for any $\ell$. Being $\xi _{\ell }$ analytic, the total transform $\sigma _0^*\xi _\ell$ exists and is analytic in a neighborhood of the divisor $E_0=\sigma _{0}^{-1}(0)$. Moreover, in terms of coordinates of the charts $C_{-\infty },\ C_0, \ C_\infty$ (cf. Sect. 3.2), we can prove (see for instance the computations in [1, sec. 3])

• (a) For $(C_\infty ,(x^{(\infty )},y^{(\infty )},z^{(\infty )}))$ (and analogously for $C_{-\infty }$) the coefficients of $\xi _{\ell }^{(\infty )}:=(\sigma _{0}|_{C_\infty })^*\xi _{\ell }$ belong to $\mathbb {R}[x^{(\infty )},y^{(\infty )}][[z^{(\infty )}]]\cap \mathbb {R}\{ x^{(\infty )},y^{(\infty )},z^{(\infty )} \}$. In fact, they belong to the algebra $\mathbb {R}[x^{(\infty )},y^{(\infty )}]\{ z^{(\infty )} \}$ of convergent series with polynomial coefficients (cf. notations at the end of Sect. 1).

• (b) For $(C_0,(\theta , z,\rho ))$, the coefficients of $\xi _{\ell }^{(0)}:=(\sigma _{0}|_{C_0})^*\xi _{\ell }$ belong to $\mathbb {R}[\cos \theta , \sin \theta , z][[\rho ]]\cap \mathbb {R}[\cos \theta , \sin \theta ]\{ z,\rho \}$. In fact, they belong to $\mathbb {R}[\cos \theta , \sin \theta ,z]\{ \rho \}$.

Finally, taking into account that $\ell \ge 1$ (i.e. $\xi _{\ell }$ has the same linear part as $\xi$ or $\hat{\xi }$), we may observe that $\xi _{\ell }^{(J)}|_{E_0\cap C_J}=\hat{\xi }^{(J)}|_{E_0\cap C_J}$ for any $J\in \{ -\infty ,0,\infty \}$. In particular, the characteristic elements of $\hat{\xi }$ in $E_0$ are invariant for the transform $\sigma _0^*\xi _{\ell }$, then $\sigma _0^*\xi _{\ell }$ admits a transform which is analytic if we blow up again one of those characteristic elements. Using recursively the same kind of arguments, and with a similar proof, we obtain the following version of Proposition 3.12 for the jets approximations of the normal form.

Proposition 3.13

Let $\mathcal {M}=(M,\pi ,\mathcal {A}, \mathcal {D})$ be an admissible sequence of blowing-ups of length l with $\mathcal {A}=\{ C_J \}_{J\in \mathcal J}$. Then, for $\ell \ge l+1$ and $J\in \mathcal J$, the transform $\xi _{\ell }^{(J)}:=(\pi |_{C_J})^*\xi _{\ell }$ is analytic. Moreover, for any $k\in \mathbb {N}$, if u is a coordinate of $C_J$ such that $\{ u= 0\}\subset E=\pi ^{-1}(0)$ and $\ell \ge k+l+1$, then, we have

$$\begin{aligned} j_k^u(\xi _\ell ^{(J)})=j_k^u(\hat{\xi }^{(J)}). \end{aligned}$$

Remark 3.14

As a part of the proof, we can see that the restriction of $\xi _{\ell }^{(J)}$ and $\hat{\xi }^{(J)}$ to the divisor coincide. Hence, the characteristic elements $\gamma _I\in \mathcal {D}$ are invariant for the total transform $\pi ^*\xi _{\ell }$. They are called characteristic singularities or characteristic cycles, accordingly, of $\xi _{\ell }^{(J)}$. Moreover, we observe that the coefficients of $\xi _{\ell }^{(J)}$ are convergent series in the coordinates of the chart $C_J$, i.e., they satisfy the corresponding property (a), respectively (b), above when $J\in \{ I_{-\infty }^\mathcal {M}, I_{\infty }^\mathcal {M} \}$ (resp. $J \notin \{ I_{-\infty }^\mathcal {M}, I_{\infty }^\mathcal {M} \}$). In the last case, we can also interchange the roles of the coordinates z and $\rho$ if $C_J$ is a corner chart.

Recall, from Sect. 3.3, the definition of the associated two dimensional vector fields $\hat{\eta }_J'$ to $\hat{\xi }^{(J)}$ for $J\in \mathcal J{\setminus } \{ I_\infty ^{\mathcal {M}},I_{-\infty }^{\mathcal {M}} \}$ and the corresponding reduced vector fields $\hat{\eta }_J'=(\rho ^{n_1^{(J)}} z^{n_2^{(J)}})^{-1} \hat{\eta }_J$, where $(\theta ,z,\rho )$ are the coordinates in $C_J$. Write the transform $\xi _\ell ^{(J)}$ as

$$\begin{aligned} \xi _\ell ^{(J)}= B_{\ell ,\theta }^{(J)}(\theta ,z, \rho ) \frac{\partial }{\partial \theta }+ B_{\ell ,z}^{(J)}(\theta ,z, \rho ) \frac{\partial }{\partial z}+B_{\ell ,\rho }^{(J)}(\theta ,z, \rho ) \frac{\partial }{\partial \rho }. \end{aligned}$$

(28)

Then, the associated (to $\xi _\ell ^{(J)}$) system of ODEs $\eta _{\ell ,J}$ is defined as:

$$\begin{aligned} \left\{ \begin{array}{lcll} \frac{d z}{d \theta } & =& B_{\ell ,\rho }^{(J)}(\theta ,z, \rho )\cdot (B_{\ell , \theta }^{(J)}(\theta ,z, \rho )) ^{-1} \\ \frac{d \rho }{d \theta } & =& B_{\ell ,z}^{(J)}(\theta ,z, \rho )\cdot (B_{\ell , \theta }^{(J)} (\theta ,z, \rho ) )^{-1} \end{array} \right. \end{aligned}$$

(29)

Recall also that, if $J\in \{ I_\infty ^{\mathcal {M}},I_{-\infty }^{\mathcal {M}} \}$ and we use simplified notation $(x,y,z):=(x^{(J)},y^{(J)},z^{(J)})$, we have defined $n^{(J)}$ as the maximum $n\in \mathbb {N}$ such that $\hat{\xi } ^{(J)} (z)$ is divisible by $z^n$. As well, if $J\in \mathcal {J}{\setminus } \{ I_\infty ^{\mathcal {M}},I_{-\infty }^{\mathcal {M}} \}$, we have defined $n^{(J)}:=\max \{ n_1^{(J)}, n_2^{(J)} \}$. With those notations, we have the following Corollary of Proposition 3.13.

Corollary 3.15

Let $\mathcal {M}=(M,\pi ,\mathcal {A}, \mathcal {D})$ be an admissible sequence of blowing-ups of length $l>0$ with $\mathcal {A}=\{ C_J \}_{J\in \mathcal J}$. Define $\ell _{\mathcal { M}}:= \max \{ n^{(J)}: J\in \mathcal {J} \} +l+1$. Fix $k\in \mathbb {N}_{\ge 0}$.

• (1) Let $J\in \mathcal {J}{\setminus } \{ I_{-\infty } ^\mathcal {M}, I_{\infty } ^\mathcal {M} \}$ and put $(z,\rho ):=(z^{(J)}, \rho ^{(J)})$. For every $\ell \ge \ell _{\mathcal { M}} + k$, the monomial $(\rho )^{n_1^{(J)}}(z )^{n_2^{(J)}}$ divides the system $\eta _{\ell ,J}$. Moreover, putting $\eta _{\ell , J}':=(\rho ^{n_1^{(J)}}z^{n_2^{(J)}})^{-1} \eta _{\ell , J}$, if u is a coordinate with $\{ u=0 \}\subset E\cap C_J$, then

  $$\begin{aligned} j^u_{k}(\eta _{\ell , J}')=j^u_{k}(\hat{\eta }_{ J}'). \end{aligned}$$

• (2) Let $J\in \{ I_{-\infty } ^\mathcal {M}, I_{\infty } ^\mathcal {M} \}$ and put $(x,y,z):=(x^{(J)},y^{(J)},z^{(J)})$. For every $\ell \ge \ell _{\mathcal { M}} + k$, the series $\xi _{\ell }^{(J)}(z)$ is divisible by $z^{n^{(J)}}$, and

  $$\begin{aligned} j^z_{k}\left( {z^{-n^{(J)}}} \xi _\ell ^{(J)}(z)\right) = j^z_{k}\left( {z^{-n^{(J)}}} \hat{\xi }^{(J)} (z)\right) . \end{aligned}$$

Proof

Both statements are direct consequence of the jet equality stated in Proposition 3.13. Since $k+\ell _{\mathcal { M}}\ge n_i^{(J)} + l +1$ for $i=1,2$ and for very $J\in \mathcal {J}{\setminus } \{ I_{-\infty } ^\mathcal {M}, I_{\infty } ^\mathcal {M} \}$ and $k+\ell _{\mathcal { M}}\ge n^{(J)}+ l+1$ when $J\in \{ I_{-\infty } ^\mathcal {M}, I_{\infty } ^\mathcal {M} \}$, we have that the monomials of type $(\rho ^{(J)})^{n_1^{(J)}}(z ^{(J)})^{n_2^{(J)}}$ divide the system $\eta _{\ell , J}$ when $J\in \mathcal {J}{\setminus } \{ I_{-\infty } ^\mathcal {M}, I_{\infty } ^\mathcal {M} \}$, or $(z ^{(J)})^{n^{(J)}}$ divides $\xi _{\ell }^{(J)}(z^{(J)})$ when $J\in \{ I_{-\infty } ^{\mathcal M}, I_{\infty } ^\mathcal {M} \}$. $\square$

3.6 Lifting of Automorphisms by Admissible Blowing-Ups

Recall from Eq. (3) that there is a formal automorphism $\psi _\ell$ at $0\in \mathbb {R}^3$ that conjugates $\hat{\xi }$ and $\xi _{\ell }$, that is, $\hat{\xi }=\psi ^*_\ell (\xi _{\ell })$, and that $\psi _\ell$ is tangent to the identity up to order $\ell$, i.e., $j_\ell (\psi _\ell -Id)=0$. In what follows, we will need to lift such a conjugation to the charts of a sequence of admissible blowing-ups. As well, we will need to lift the (analytic) conjugation between different jet approximations $\xi _{\ell }$ and $\xi _{\ell '}$. We provide a proper statement covering all those situations.

Proposition 3.16

Let $\mathcal {M}=(M,\pi ,\mathcal {A}, \mathcal {D})$ be a sequence of admissible blowing-ups of length l. Take $\ell >l+1$ and let $\psi \in \mathbb {R}[[x,y,z]]$ be a formal automorphism satisfying $j_{\ell }(\psi -(x,y,z))=0$. Then, for any $C_{J}\in \mathcal {A}$, there exists a formal automorphism $\psi ^{(J)}$ in the coordinates of $C_J$ satisfying

$$\begin{aligned} \pi |_{C_J} \circ \psi ^{(J)}=\psi \circ \pi |_{C_J}. \end{aligned}$$

(30)

More precisely, if $\ell =k+l+1$, we have:

• (1) Suppose $J=I^\mathcal { M}_{\pm \infty }$ and denote $(u,v,w):=(x^{(J)},y^{(J)},z^{(J)})$, then there is $\psi ^{(J)}\in \mathbb {R}[u,v] [[w]]^3$ that satisfies $j^w_k(\psi ^{(J)}- (u,v,w))=0$.

• (2) Suppose $J\ne I^\mathcal { M}_{\pm \infty }$ and denote $(\theta , z,\rho ):=(\theta , z^{(J)}, \rho ^{(J)})$ (notice that $\{ \rho =0 \}\subset E\cap C_J$). Then, $\psi ^{(J)}=\psi ^{(J)}(\theta ,z,\rho )=(\theta + F_\theta , z+F_z,\rho + F_\rho )$ where each $F_i\in \mathbb {R}[\cos \theta , \sin \theta , z][[\rho ]]$ and $\psi ^{(J)}$ satisfies $j_{k}^{\rho }(\psi ^{(J)}-(\theta ,z,\rho ))=0$. Moreover, if $\{ z=0 \}\subset E\cap C_J$ ($C_J$ is a corner chart), then we have also $F_i\in \mathbb {R}[\cos \theta ,\sin \theta ,\rho ][[z]]$.

• (3) In the same conditions, assume moreover that $\psi \in \mathbb {R}\{x,y,z\}^3$ is convergent. Then, $\psi ^{(J)}\in \mathbb {R}[u,v]\{ w\}$ in case (1) and $\psi ^{(J)}\in \mathbb {R}[\cos \theta , \sin \theta ,z]\{ \rho \}$ in case (2).

Proof

We can write $\pi =\sigma _0\circ \sigma _{1} \circ \cdots \circ \sigma _{r'} \circ \cdots \circ \sigma _{r}$ with $\sigma _i=\sigma _{\gamma _{I_i}}$ and $0\le r'\le r$, where $\gamma _{I_i}$ are characteristic singularities of the form $\gamma _{(\varepsilon \infty ,{\mathop {\ldots }\limits ^{m_i}},\varepsilon \infty )}$ for $0\le i \le r'$ and $\varepsilon =\pm 1$, and $\gamma _{I_{j}}$ are characteristic cycles for $1+r'\le j\le r$.

Suppose that $r'\ge 1$. First, the automorphism $\psi$ can be lifted to $\psi ^{(\infty )}$ at the point $\gamma _\infty$ (or, correspondingly, to $\psi ^{(-\infty )}$ at the point $\gamma _{-\infty }$): using the chart $(C_\infty , (x^{(\infty )}, y^{(\infty )}, z^{(\infty )} ))$ and the quadratic expression of $\sigma _0|_{C_\infty }$ given in (5), the formal automorphism defined by

$$\begin{aligned} \psi ^{(\infty )} (x^{(\infty )}, y^{(\infty )}, z^{(\infty )} )= \left( \frac{x\circ \psi }{z\circ \psi }, \frac{y\circ \psi }{z\circ \psi },z\circ \psi \right) \circ \sigma _0|_{C_\infty } (x^{(\infty )}, y^{(\infty )}, z^{(\infty )} ) \end{aligned}$$

(31)

satisfies that $\sigma _0 |_{C_\infty } \circ \psi ^{(\infty )}= \psi \circ \sigma _0 |_{C_\infty }$. Moreover, using that $j_{\ell }(\psi )=Id$ and the explicit expression of $\psi ^{(\infty )}$, we get that $j_{\ell -1}(\psi ^{(\infty )})=Id$. In addition, it is standard to prove from (31) that $\psi ^{(\infty )}$ belongs to $\mathbb {R}[x^{(\infty )},y^{(\infty )}][[z^{(\infty )}]]$, and more precisely to $\mathbb {R}[x^{(\infty )},y^{(\infty )}]\{z^{(\infty )}\}$ when $\psi$ is convergent. Thus, $\psi ^{(J)}$ satisfies the required properties (1) and (3) for $J\in \{ -\infty ,\infty \}$ when $r=1$. Moreover, $\psi ^{(\infty )}$ satisfies at $\gamma _\infty$ the same properties as $\psi$ does at 0, renaming $\ell :=\ell -1$. Repeating the same arguments for each blowing-up in the composition $\pi _1=\sigma _0\circ \cdots \sigma _{r'-1}$ when $r'\ge 1$, we obtain that there is a formal automorphism $\psi ^{I_{r'}}$ at $\gamma _{I_{r'}}$ such that $\pi |_{C_{I_{r'}}}\circ \psi ^{I_{r'}}=\psi \circ \pi |_{C_{I_{r'}}}$ and satisfying the same hypothesis as $\psi$ at 0, but putting $\ell -r'$ instead of $\ell$. Thus, for $J=I_{-\infty }^\mathcal {M},I_{\infty }^\mathcal {M}$, we have shown items (1) and (3). Renaming the point $\gamma _{I_{r'}}$ as the origin when $r'\ge 1$, we can assume that $r'=0$. Let us analyze the chart $(C_0,(\theta ,z,\rho ))$ of the blowing-up $\sigma _0$. Write $\psi (x,y,z)=(x+G_1,y+G_2,z+G_3)$ where each $G_i\in \mathbb {R}[[x,y,z]]$ has order at least $\ell +1$. We have

$$\begin{aligned} \psi \circ \sigma _0|_{C_0}(\theta , z,\rho )= & (\rho \cos \theta + G_1(\rho \cos \theta , \rho \sin \theta ,\rho z),\rho \sin \theta + G_2(\rho \sin \theta , \rho \sin \theta ,\rho z),\nonumber \\ & \rho z + G_3(\rho \cos \theta , \rho \sin \theta ,\rho z)). \end{aligned}$$

(32)

Hence, $G_i\circ \sigma _0|_{C_0} \in \mathbb {R}[\cos \theta , \sin \theta ,z ][[ \rho ]]$ and $\rho ^{\ell +1}$ divides each $G_i\circ \sigma _0|_{C_0}$. Moreover, these series $G_i\circ \sigma _0|_{C_0}$ belong to $\mathbb {R}[\cos \theta , \sin \theta ,z]\{\rho \}$ if $\psi$ is convergent. We introduce $\tilde{G}_i=\frac{1}{\rho }G_i\circ \sigma _0|_{C_0}$. We look for a formal automorphism of the form

$$\begin{aligned} \psi ^{(0)}(\theta , z,\rho )=(\theta + \rho F_1(\theta , z,\rho ), z+\rho F_2(\theta , z,\rho ),\rho + \rho F_3(\theta , z,\rho )) \end{aligned}$$

such that $\sigma _0|_{C_0}\circ \psi ^{(0)}=\psi \circ \sigma _0|_{C_0}$. The tuple $(F_1,F_2,F_3)$ must fulfill

$$\begin{aligned} \begin{aligned} \rho \cos \theta + \rho {\tilde{G}}_1&=(\rho + \rho F_3 ) \cos (\theta + \rho F_1), \\ \rho \sin \theta +\rho \tilde{G}_2&=(\rho + \rho F_3) \sin (\theta +\rho F_1), \\ \rho z + \rho {\tilde{G}}_3&=(z + \rho F_2 ) (\rho +\rho F_3 ). \end{aligned} \end{aligned}$$

Put ${\tilde{F}}_1:=\rho F_1$ and ${\tilde{F}}_2=\rho F_2$. Using classical formulas for trigonometric functions, and dividing the above expressions by $\rho$, we obtain the following system of formal equations with coefficients in the ring $\mathbb {R}[\cos \theta , \sin \theta ,z,\rho ]$ and in the variables $({\tilde{G}},{\tilde{F}})=({\tilde{G}}_1,{\tilde{G}}_2,{\tilde{G}}_3, {\tilde{F}}_1, {\tilde{F}}_2, F_3)$.

$$\begin{aligned} \begin{aligned} {\tilde{G}}_1&= \cos \theta F_3- \sin \theta {\tilde{F}}_1- \sin \theta F_3 {\tilde{F}}_1 + O({\tilde{F}}_1^ 2), \\ {\tilde{G}}_2&= \sin \theta F_3+ \cos \theta {\tilde{F}}_1+ \cos \theta F_3 {\tilde{F}}_1 + O({\tilde{F}}_1^ 2), \\ \tilde{G}_3&= z F_3 +\tilde{F}_2 + F_3\tilde{F}_2. \end{aligned} \end{aligned}$$

(33)

The differential of the system with respect to the unknown variables ${\tilde{F}}$ at ${\tilde{F}}=0$ is invertible, as a matrix with entries in $\mathbb {R}[\cos \theta , \sin \theta ,z,\rho ]$. We apply the implicit function theorem to find a solution ${\tilde{F}}\in \mathbb {R}[\cos \theta , \sin \theta ,z,\rho ][[{\tilde{G}}]]^3$, see for example [4, A.IV.37]. Notice also that ${\tilde{G}}\in \mathbb {R}[\cos \theta , \sin \theta ,z][[\rho ]]$, and hence ${\tilde{F}}\in \mathbb {R}[\cos \theta , \sin \theta ,z][[\rho ]]^3$. Since $j_{\ell -1}^\rho ({\tilde{G}}_i)=0$ for $i=1,2,3$, we find that $\rho ^{\ell }$ divides $\rho F_1, \rho F_2,F_3$ and $j_{\ell -1}(\psi ^{(0)})=Id$. Once more, if $\psi$ is convergent, then we get that $\psi ^{(0)}\in \mathbb {R}[\cos \theta , \sin \theta ,z]\{ \rho \}^3$ since $G_i\in \mathbb {R}[\cos \theta , \sin \theta ,z]\{ \rho \}$. (Notice that the system (33) is composed by equations that belong to $\mathbb {R}[\cos \theta , \sin \theta ,z,\rho ]\{ {\tilde{G}},{\tilde{F}}\}$, in this case). We get the desired lifting $\psi ^{(0)}$ of $\psi$ in the chart $C_0$.

We proceed studying the rest of the blowing-ups $\sigma _i, \ i\ge 1$, by recurrence. Let us simply discuss the first step of the recurrence, the rest is done similarly. Up to a translation $z\mapsto z+\omega$ in $\{\rho =0\}$, the expression of $\sigma _1$ (omitting super-indices) is either $\sigma _{1}(\theta ,z, \rho )=(\theta ,\rho z, \rho )$ (in the non-corner chart $C_{J_0}$ as in Eq. (21)) or $\sigma _1(\theta , z, \rho )=(\theta , \pm z, z \rho )$ (in a corner chart, say $C_{J_{\pm \infty }}$, as in Eqs. (20) or (22)). We obtain the desired expression of $\psi ^{(J)}$ proceeding as in (31). For example, in the situation of the corner chart $J=J_\infty$, we define

$$\begin{aligned} \psi ^{(J)} (\theta ,z,\rho ):=\left( \theta \circ \psi ^{(0)},z\circ \psi ^{(0)},\frac{\rho \circ \psi ^{(0)}}{z\circ \psi ^{(0)}} \right) \circ \sigma _{1}|_{C_J}. \end{aligned}$$

Considering the expression of $\sigma _{1}|_{C_J}$, the coefficients of $\psi ^{(J)}$ belong to $\mathbb {R}[\cos \theta , \sin \theta , \rho ][[z]]$ and also to $\mathbb {R}[\cos \theta , \sin \theta , z][[\rho ]]$ since the coefficients of $\psi ^{(0)}$ belong to $\mathbb {R}[\cos \theta , \sin \theta , z][[\rho ]]$. By the fact that $j_{\ell -1}^\rho ( \psi ^{(0)} - (\theta ,z,\rho ) )=0$ and the above expression, we deduce $j_{\ell -2}^u(\psi ^{(J)}-(\theta ,z,\rho ))=0$ for $u=\rho ,z$. This shows (2) for $J=J_\infty$. To prove (3) for this same index, we observe that through all the operations made above (including the translation in z), the convergent nature of $\psi ^{(J_\infty )}$ is inherited from that of $\psi ^{(0)}$. $\square$

4 Characteristic Cycles as Limit Sets

In this section, we use the analytic approximations $\xi _\ell$ to the formal normal form $\hat{\xi }$ with an objective: proving that the characteristic elements of $\xi _\ell$ after a sequence of admissible blowing-ups $\mathcal M$ are the only possible limit sets of the family of local cycles of $\xi _\ell$ for $\ell$ large enough.

Along this section, we fix a sequence of admissible blowing-ups $\mathcal {M}=(M,\pi ,\mathcal {A},\mathcal {D})$ for $\hat{\xi }$, with $\mathcal {A}=\{ C_J \}_{J\in \mathcal J}$ and $\mathcal {D}=\{ \gamma _I \}_{I\in \mathcal I}$. Denote by $E=\pi ^{-1}(0)$ the total divisor of $\pi$. We define also the support of $\mathcal {D}$ as $\text {Supp}\mathcal {D}=\displaystyle \bigcup _{I\in \mathcal I} \gamma _I$. Recall the definition of $\ell _{\mathcal { M}}$ in Corollary 3.15.

Proposition 4.1

Let $\ell \ge \ell _{\mathcal { M}}+1$ and W be a neighborhood of $\text {Supp}\mathcal {D}=\displaystyle \bigcup _{I\in \mathcal I} \gamma _I$. There is some neighborhood $U=U(W)$ of $0\in \mathbb {R}^3$ such that $\pi ^{-1}(\mathcal C_U(\xi _ \ell ))\subseteq W$.

To prove this result, we need to introduce new notation and a technical lemma. Consider the set

$$\begin{aligned} \dot{E}:=E{\setminus } \left( \left( \displaystyle \bigcup _{I:\gamma _I\in \mathcal {D} \text { corner}} \{\gamma _I \} \right) \cup \{\gamma _{I_\infty ^\mathcal {M}}\} \cup \{\gamma _{I_{-\infty }^\mathcal {M}} \} \right) . \end{aligned}$$

The set $\dot{E}$ has a finite family of connected components denoted by $\mathcal {E}_{\mathcal {M}}= \{ L_0, L_1, \ldots , L_{k_\mathcal { M}}\}$. Each $L_i\in \mathcal {E}_{\mathcal { M}}$ is open in E and contained in a chart $C_{J_i}$ for $i=0,1,\ldots , k_\mathcal {M}$. Therefore, we will call them simply open components (of E). In addition, in case $L_i$ is contained in two different charts, we choose $C_{J_i}$ such that $L_i\subseteq \{ \rho ^{(J_i)}=0 \}$, which is always possible by the hypothesis (H3) of sequences of admissible blowing-ups. Then, each open component $L_i= \mathbb {S}^1\times (\lambda _i^-,\lambda ^+_i)\times \{0\}$ in the coordinates of $C_{J_i}$ where $\lambda _i^-\in \mathbb {R}\cup \{-\infty \}$ and $\lambda _i^+\in \mathbb {R}\cup \{\infty \}$. An element $L_i\in \mathcal {E_M}$ is said to be dicritical (respectively, non-dicritical) if the component of E that contains $L_i$ is dicritical (respectively, non-dicritical).

Fix $L=L_i\in \mathcal {E_M}$, with $L=\mathbb {S}^1\times (\lambda ^-,\lambda ^+) \times \{0\}$, and the corresponding chart $C_{J}$ with $J=J_i$. Consider the formal vector field $\hat{\eta }_J$ associated to $\hat{\xi }^{(J)}=(\pi |_{C_{J}})^*\hat{\xi }$ as in Eq. (11). For the purpose of this section, we write, removing super-indices in $(z,\rho )$:

$$\begin{aligned} \hat{\eta }_ J= \rho ^{n_1^{(J)}} ( A^{(J)}_z(z,\rho )\frac{\partial }{\partial z}+ A^{(J)}_\rho (z,\rho )\frac{\partial }{\partial \rho }). \end{aligned}$$

(34)

Notice that there is a small modification here with respect to Eq. (11): we include the factor $z^{n_2^{(J)}}$ in the coefficients $A_j^{(J)}$ for $j=z,\rho$ (which may be non-trivial if $C_J$ is a corner chart) in Eq. (34). The reduced vector field $\hat{\eta }_J'':=\rho ^{-n_1^{(J)}}\hat{\eta }_J$ considered here (not necessarily equal to $\hat{\eta }_J'$) has a finite number of points in the adapted singular locus of $\hat{\eta }_J'$ relatively to $F=E\cap \{\theta = 0\}$ along $\{ \rho =0 \}$, which determine the characteristic cycles contained in L. The $z-$coordinates of the characteristic cycles in L are denoted by $\omega _1^{L}, \ldots , \omega ^L_{m_{L}}$ and the associated characteristic cycles by $\gamma ^L_{1}, \ldots , \gamma ^L_{m_L}$.

Define the collection of sets $\mathcal V (L,\varepsilon ,\delta ):= \{ V_0, V_1,\ldots , V_{m_{L}-1}, V_{m_{L}} \}$ depending on two parameters $\varepsilon ,\delta >0$ by:

$$\begin{aligned} V_{0} & = \mathbb {S}^ 1 \times \Omega _0(\varepsilon ) \times (0,\delta ], \quad \Omega _0(\varepsilon )= [\mu _-, \omega _1^{L}-\varepsilon ], \nonumber \\ V_{j} & =\mathbb {S}^ 1 \times \Omega _j(\varepsilon ) \times (0,\delta ], \quad \Omega _j(\varepsilon )=[\omega _{j}^{L}+\varepsilon , \omega _{j+1}^{L}-\varepsilon ], \quad j=1, \ldots , m_{L}-1, \nonumber \\ V_{m_{L}} & =\mathbb {S}^ 1 \times \Omega _{m_L}(\varepsilon ) \times (0,\delta ], \quad \Omega _{m_L}(\varepsilon )=[\omega _{m_L}^{L}+\varepsilon , \mu _+], \end{aligned}$$

(35)

where $\mu _\pm =\lambda ^\pm \mp \varepsilon$ when $|\lambda ^\pm |<\infty$, $\mu _- =\omega _{1}^{L}-\frac{1}{\varepsilon }$ when $\lambda ^-=-\infty$, and $\mu _+=\omega _{m_L}^L+\frac{1}{\varepsilon }$ when $\lambda ^+=\infty$. Define the surfaces $\partial _{min}V_j$ and $\partial _{max}V_j$ as follows:

• $\partial _{min}V_0= \mathbb {S}^1 \times \{ \mu _- \} \times (0,\delta ]$ and $\partial _{min} V_j= \mathbb {S}^1 \times \{ \omega _{j}^{L}+\varepsilon \} \times (0,\delta ]$ for $j=1,2,\ldots ,m_{L}$.

• $\partial _{max}V_j= \mathbb {S}^1 \times \{ \omega _{j+1}^{L}-\varepsilon \} \times (0,\delta ]$ for $j=0,1,\ldots ,m_{L}-1$ and $\partial _{max}V_{m_{L}}= \mathbb {S}^1 \times \{ \mu _+ \} \times (0,\delta ]$.

Notice that elements of the family $\mathcal {V}(L,\varepsilon ,\delta )$ are subsets of the corresponding elements of $\mathcal {V}(L,\varepsilon ',\delta )$ for any $\varepsilon '<\varepsilon$.

Lemma 4.2

Assume $\ell \ge \ell _{\mathcal { M}}+1$ and denote by $\xi _\ell ^{(J)}=(\pi |_{C_J})^*\xi _\ell$. There exists $\varepsilon _0>0$ such that for every small $\varepsilon$ with $\varepsilon _0>\varepsilon >0$, there exists $\delta =\delta (\varepsilon )>0$ such that the collection $\mathcal V (L,\varepsilon ,\delta )= \{ V_j \}_{j=0}^{m_L}$ satisfies:

• (1) In case L is non-dicritical, the function z is monotonic along the trajectories of $\xi _\ell ^{(J)}$ in each $V_j$ for any j. Otherwise, if L is dicritical, the function $\rho$ is monotonic along the trajectories of $\xi _\ell ^{(J)}$ in each $V_j$, for any j.

• (2) If L is dicritical and $\rho ^{n_1^{(J)}+1}$ does not divide $\xi _\ell ^{(J)}(z)$, then $\xi _\ell ^{(J)}(z)$ has constant sign along the surfaces $\partial _{min}V_j$ and $\partial _{max}V_j$, for any j.

• (3) Suppose that L is dicritical and $\rho ^{n_1^{(J)}+1}$ divides $\xi _\ell ^{(J)}(z)$. Denote $\mathcal V(L,\frac{\varepsilon }{2}, \delta )=\{ V_0',V_1',\ldots , V_{m_{L}}' \}$. Then, each element $V'_j\in \mathcal V(L,\frac{\varepsilon }{2}, \delta )$ fulfills (1) and, moreover, any trajectory of $\pi ^*\xi _\ell$ containing a point in $V_j$ remains inside $V_j'$ either for any positive time $t\ge 0$ or for any negative time $t\le 0$.

Proof

Taking into account Corollary 3.15 and since $\ell \ge \ell _{\mathcal { M}}$, the vector field $\xi _\ell ^{(J)}$ is described by a non-autonomous two dimensional system of ODEs (see Eq. (29))

$$\begin{aligned} \left\{ \begin{array}{lcll} \frac{d z}{d \theta } & =& \rho ^{n_1^{(J)}} A^{\ell ,(J)}_z (\theta ,z, \rho ), \\ \frac{d \rho }{d \theta } & =& \rho ^{n_1^{(J)}} A^{\ell ,(J)}_\rho (\theta ,z, \rho ) \end{array} \right. \end{aligned}$$

(36)

where $A^{\ell ,(J)}_u(\theta ,z, 0)=A^{(J)}_u(z,0)$ for $u=\rho ,z.$ (As for the formal system of ODEs (34), we include the factor $z^{n^{(J)}_2}$ in $A^{\ell ,(I)}_u$).

We choose $\varepsilon _0$ satisfying the following conditions:

• In any case, we require $\varepsilon _0< \tfrac{1}{2}\min _{i\ne j} \{ |\omega _i^L-\omega _j^L| \}$.

• When L is dicritical, if we have that ${A_z^{\ell ,(I)}}(\theta ,z,0)\not \equiv 0$ and $\{t_1,\ldots , t_s\}$ is its set of zeroes, in order to prove property (2), we require also

  $$\begin{aligned} \varepsilon _0< \frac{1}{2} \displaystyle \min \{ |\omega _j^L-t_{k}|\,\ {1\le j \le m_L,\ 1\le k \le s, \ \omega ^L_j\ne t_{k} } \}. \end{aligned}$$

In the non-dicritical case, the function $A_z^{\ell ,(J)}(\theta ,z,0)\equiv A_z^{(J)}(z,0)$ is not identically zero and only depends on z. Being its zeroes $\omega _1^{L}, \ldots , \omega ^L_{m_{L}}$ by definition, it has constant sign when z belongs to the interval of $\Omega _j(\varepsilon )$ for $j\in \{0,\ldots ,m_L\}$ for any $0<\varepsilon <\varepsilon _0$. By continuity and periodicity in $\theta$, $A_z^{\ell ,(J)}(\theta ,z,\rho )$ has constant sign for $(\theta ,z,\rho )$ in a set of the form $\mathbb {S}^1\times \Omega _j(\varepsilon )\times (0,\delta _j]$ for some $\delta _j=\delta _j(\varepsilon )$. Take $\delta$ fulfilling $\delta \le \displaystyle \min _{i=0, \ldots , m_{L_i}} \{ \delta _i \}$ and $B_{\ell ,\theta }^{(J)}=\xi _\ell ^{(J)}(\theta )$ has positive sign in $\mathbb {S}^1\times \Omega _j(\varepsilon )\times (0,\delta ]$ for every $j=0, \ldots , m_{L}$. This is possible since $B_{\ell ,\theta }^{(J)}(\theta ,0,0)=1$. Then, we define $V_j:=\mathbb {S}^1 \times \Omega _j(\varepsilon ) \times (0,\delta ]$. Taking into account that $\xi _{\ell }^{(J)}(z)=\rho ^{n_1^{(J)}} A^{\ell ,(J)}_z(\theta ,z,\rho ) \cdot B_{\ell ,\theta }^{(J)}( \theta ,z, \rho )$, we obtain the property (1) for the non-dicritical case.

In the dicritical case we proceed in the same way. Notice that $A_\rho ^{\ell ,(J)}(\theta ,z,0)=A_\rho ^{(J)}(z,0)$ only depends on z and its set of zeros is by definition $\omega _1^L,\ldots ,\omega _{m_L}^L$. We get that $\xi _\ell ^{(J)}(\rho )$ has constant sign in each $V_j$ and statement (1) holds.

Let us show (2), assuming that $\rho ^{n_1^{(J)}+1}$ does not divide $\xi _\ell ^{(J)}(z)$. By the choice of $\varepsilon _0$, we have that $A_z^{\ell ,(J)}(\theta ,z,0)= A^{(J)}_z(z,0)$ does not vanish at any of the extreme values of $\Omega _j(\varepsilon )$. Since $\xi ^{(J)}_\ell (z)=\rho ^{n_1^{(J)}} A^{(J)}_z(\theta ,z,\rho ) \cdot B_{\ell ,\theta }^{(J)}( \theta ,z, \rho )$, we obtain (2), up to taking a smaller $\delta$.

Finally, we show (3). Assume that L is dicritical and that $A_z^{\ell ,(J)}(\theta ,z,0)\equiv 0$. Then, the system (36) associated to $\xi ^{(J)}_\ell$ can be written as

$$\begin{aligned} \left\{ \begin{array}{lll} \frac{dz}{d\theta }& = & \rho ^{n^{(J)}_1+1}{\widetilde{A}}_{z} ^{\ell ,(J)}(\theta , z,\rho )\\ \frac{d\rho }{d\theta }& = & \rho ^{n^{(J)}_1}A_{\rho } ^{\ell ,(J)}(\theta , z,\rho ) \end{array} \right. , \end{aligned}$$

(37)

where $A_{\rho } ^{\ell ,(J)}(\theta , z,0 )$ does not depend on $\theta$ (by Corollary 3.15), vanishes exactly for $z\in \{ \omega _1^L,\ldots ,\omega _{m_L}^L \}$, and $\widetilde{A}_{z} ^{\ell ,(J)}( \theta ,z,0 )\in \mathbb {R}[\cos \theta ,\sin \theta ,z]$. Proceeding as in the beginning of the proof, we take a constant $\delta >0$ such that the collection $\mathcal {V}(L,\frac{\varepsilon }{2}, \delta )=\{ V_0',V_1',\ldots , V_{m_L}' \}$ fulfills (1), so that $\rho$ is monotonic in every $V_j'$. Being $\overline{V}_j$ compact, there are constants $a,K>0$ such that for any $V_j'\in \mathcal {V}(L,\frac{\varepsilon }{2}, \delta )$, we have

$$\begin{aligned} \inf _{p\in V_j'} \{ |A_{\rho } ^{\ell ,(J)}(p) | \} \ge a, \quad \sup _{p\in V_j'} \{ |{\widetilde{A}}_{z} ^{\ell ,(J)}(p) | \} \le K. \end{aligned}$$

(38)

Fix $V_j'$ and suppose, for instance, that $A_{\rho } ^{\ell ,(J)}|_{V_j'}<0$. Then, if $\sigma :\mathbb {R}\longrightarrow M$ is a trajectory of $\xi _\ell ^{(J)}$ parameterized as a solution $\sigma (\theta )=(\theta , z(\theta ),\rho (\theta ))$ of system (37), as long as it remains in $V_j'{\setminus } L$, the function $\rho (\theta )$ is strictly decreasing. Hence, $\sigma$ can be parameterized by $\rho$ instead of $\theta$ and we obtain from (37) and (38) that

$$\begin{aligned} \left| \frac{dz}{d\rho } \right| \le C\rho , \text { where } C=\frac{K}{a}. \end{aligned}$$

Now, consider the collection $\mathcal {V}(L,\varepsilon , \delta )=\{ V_0,V_1,\ldots , V_{m_L} \}$ whose elements fulfill $V_j\subset V_j'$ for $j=0,1,\ldots , m_L$. If the trajectory $\sigma$ starts at a point $p_0=(\theta _0,z_0,\rho _0)\in V_j\subset V_j'$ with $\rho _0>0$, it satisfies, for $\theta >\theta _0$:

$$\begin{aligned} |z(\theta )-z_0|\le \frac{C}{2} |\rho (\theta )^2-\rho _0^2|\le \frac{C}{2}\rho _0^2 \le \frac{C}{2} \delta ^2 \end{aligned}$$

as long as $\text {Im}(\sigma |_{[\theta _0,\theta ]})\subset V_j'$. We obtain similar bounds for $|z(\theta )-z_0|$ when $A_{\rho } ^{\ell ,(J)}|_{V_j'}>0$. Imposing $\delta <\sqrt{\frac{\varepsilon }{C} }$, we can conclude that $|z_0-z(\theta )|<\frac{\varepsilon }{2}$ and guarantee, for any j and for any $p_0\in V_j\in \mathcal V(L,\varepsilon ,\delta )$, that the trajectory $\sigma$ starting at $p_0$ satisfies $\text {Im}(\sigma |_{[\theta _0,\infty )})\subset V_j'$ (or $\text {Im}(\sigma |_{(-\infty ,\theta _0]})\subset V_j'$ in case $A_{\rho } ^{\ell ,(J)}|_{V_j'}>0$). $\square$

From the proof above, we may observe that $\mathcal V (L,\varepsilon ,\delta ')$ also fulfills (1–3) of the lemma for any $\delta '<\delta$.

Notation 4.3

Given an open component $L\in \mathcal {E_M}$ as above, with the notations of Sect. 3.3 for $j\in \{0,\ldots ,m_L\}$, let $I_j\in \mathcal {I}$ be the index of the corresponding characteristic cycle $\gamma _{I_j}=\{ z=\omega _j, \rho =0 \}$. Let $I_0, I_{m_L+1}$ be also the indices of, either the corner characteristic cycles or characteristic singularities in the component ${\bar{L}}$. We say that the box $V_j\in \mathcal V(L,\varepsilon , \delta )$ with $j=1,\ldots , m_L-1$ is adjacent to $\gamma _{I_j}$ and to $\gamma _{I_{j+1}}$ and we denote $\partial _{I_j}V_j=\partial _{min}V_j$ and $\partial _{I_{j+1}}V_j=\partial _{max}V_j$.

Proof of Proposition 4.1

Let W be a neighborhood of $\text {Supp}\mathcal {D}$. For every $I\in \mathcal I$, we consider an open neighborhood $W_I\subset W$ of $\gamma _I$ such that $W_I\cap W_{I'}=\emptyset$ if $I\ne I'$. Consider the collection $\mathcal E_\mathcal {M}$, and apply Lemma 4.2 to each $L_i\in \mathcal E_\mathcal {M}$, taking $\varepsilon$ and $\delta$ small enough so that each family $\mathcal V (L_i,\varepsilon ,\delta )$ also satisfies:

• For any $V\in \mathcal {V}(L_i,\varepsilon , \delta )$, we impose $V\cap W_I\ne \emptyset$ if and only if $\gamma _I$ is adjacent to V.

• For any $V\in \mathcal {V}(L_i,\varepsilon , \delta )$, the boundaries $\partial _{min}V$ and $\partial _{max} V$ are contained in the corresponding neighborhoods $W_{I}$ and $W_{I'}$, where $\gamma _{I}$ and $\gamma _{I'}$ are adjacent to V.

• The set

  $$\begin{aligned} \displaystyle \bigcup _{I\in \mathcal I} W_I \cup \bigcup _{L\in \mathcal E_\mathcal {M}} \bigcup _{V\in \mathcal V(L,\varepsilon ,\delta )} V \end{aligned}$$

  is a neighborhood of the divisor $E=\pi ^{-1}(0)$ in M.

Now, we define a closed neighborhood ${{\widetilde{W}}}_I\subset W_I$ of $\gamma _I$ for each $I\in \mathcal {I}$ in such a way that (see Fig. 4):

• (i) The set

  $$\begin{aligned} {{\widetilde{U}}}=\text {int}\left( \displaystyle \bigcup _{I\in \mathcal I} {\widetilde{W}}_I \cup \bigcup _{L\in \mathcal E_\mathcal {M}} \bigcup _{V\in \mathcal V(L,\varepsilon ,\delta )} V\right) \end{aligned}$$

  is a neighborhood of the divisor E in M.

• (ii) For any $I\in \mathcal {I}$, $L\in \mathcal {E}_{\mathcal { M}}$ and $V\in \mathcal {V}(L,\varepsilon , \delta )$, ${{\widetilde{W}}}_I\cap {\bar{V}}$ is empty, in case V is not adjacent to $\gamma _I$, or, otherwise, it is of the form ${{\widetilde{W}}}_I\cap V=\mathbb {S}^1\times \{c\}\times (0,\mu ]$, where $0<\mu \le \delta$ and $c=c(V,I)$ satisfies $\partial _I V=\mathbb {S}^1\times \{c\}\times (0,\delta ]$.

Now, the set $U:=\pi ({\tilde{U}})$ is an open neighborhood of 0 satisfying the requirements of the proposition. More precisely, we claim that $\pi ^{-1} (\mathcal C_U(\xi _\ell ))\subset \displaystyle \bigcup _{I\in \mathcal I} {\widetilde{W}}_I$.

Fig. 4

Cross-section of the neighborhoods ${\widetilde{W}}_{I_j}\subset W_{I_j}$ and of $\tilde{U}$

To prove this, suppose that there is a cycle Z of $\xi _\ell$ contained in U and such that ${\widetilde{Z}}:= \pi ^{-1}(Z)$ intersects some $V\in \mathcal V(L, \varepsilon , \delta )$ for some L. Consider a parametrization $\sigma :\mathbb {R}\longrightarrow {\tilde{U}}$ of ${\widetilde{Z}}$ as a trajectory of $\pi ^*\xi _\ell$ such that $\sigma (0)\in V$. By the property (1) of Lemma 4.2, one of the coordinates z or $\rho$ is monotonic along $\sigma$ inside V, so it cannot be completely contained in V. As a consequence, $\sigma$ leaves V so that for some $t_0\ge 0$ we have $\sigma (t_0)\in \text {Fr}(V)\cap {{\widetilde{W}}}_I$, where $I\in \mathcal {I}$ and $\gamma _I$ is adjacent to V. By construction (cf. item (ii) above), $\sigma (t_0)$ belongs to the boundary $\partial _I V$. We have two cases to consider (we take notations as in Lemma 4.2).

• $A_z^{\ell , (J)}(\theta ,z,0)\ne 0$. By statement (2) of Lemma 4.2, the vector field $\pi ^*\xi _\ell$ is transverse to $\partial _I V$, so that, for instance, we have $\sigma ((t_0-c,t_0))\subset \text {int}(V)$ and $\sigma ((t_0,t_0+c))\subset \text {ext}(V)$ for some $c>0$. Since $\sigma$ is periodic, we must have that $\sigma$ crosses $\text {Fr}(V)$ at a first time $t_1>t_0$ necessarily along one of the boundaries $\partial _{min}V, \partial _{max}V$ where $\pi ^*\xi _\ell$ points towards $\text {int}(V).$ If we denote $\{ \partial _I V,\partial _{I'} V\}= \{ \partial _{min} V,\partial _{max} V\}$, we must have $\sigma (t_0)\in \partial _I V$, $\sigma (t_1)\in \partial _{I'} V$ and $\sigma ((t_0,t_1))\subset \text {ext}(V)$. Now, by construction, ${\tilde{U}}{\setminus } V={\widetilde{U}}_1 \cup {\widetilde{U}}_2$, where ${\tilde{U}}_1,{\tilde{U}}_2$ are non-empty open sets such that ${\tilde{U}}_1\cap {\tilde{U}}_2=\emptyset$ and the closure of each ${\tilde{U}}_i$ cuts V only along exactly one of the sets $\{\partial _{I'}V, \partial _{I} V\}$. We get the desired contradiction: $\sigma ((t_0,t_1))$, being connected, is contained either in ${\tilde{U}}_1$ or in ${\tilde{U}}_2$ and the extremities $\sigma (t_0)$, $\sigma (t_1)$ should belong to the same set among $\partial _{I}V, \partial _{I'}V$.

• $A_z^{\ell , (J)}(\theta ,z,0)\equiv 0$. Using statement (3) of Lemma 4.2, we know that either $\sigma ((t_0,\infty ))$ or $\sigma ((-\infty , t_0))$ is contained in the corresponding element $V'$ of the collection $\mathcal {V}(L, \frac{\varepsilon }{2},\delta )$ and $\rho \circ \sigma$ is monotonic along that interval. This is also a contradiction with $\sigma$ being periodic.

Consequently, we have proved that ${\tilde{Z}}\subset \displaystyle \bigcup _{I\in \mathcal I} {{\widetilde{W}}}_I$ (in fact, included in a single ${{\widetilde{W}}}_I$ by connectedness). Therefore, we have that:

$$\begin{aligned} \pi ^{-1}(\mathcal {C}_U(\xi _{\ell }))\subset \bigcup _{I\in \mathcal I}{\widetilde{W}}_I \subseteq \bigcup _{I\in \mathcal I} W_I\subseteq W, \end{aligned}$$

as we wanted to prove. $\square$

5 Analysis of Final Adapted Simple Singularities

Along this section, we consider some $\xi \in \mathcal {H}^3$ with fixed singularity. We fix a formal normal form $\hat{\xi }$ of $\xi$ and an adapted resolution of singularities $\mathcal { M}=(M,\pi ,\mathcal {A}, \mathcal {D})$ of $\hat{\xi }$ according to Proposition 3.10. Denote by $E=\pi ^{-1}(0)$ the exceptional divisor of $\pi$.

5.1 Infinitely Near Points of the Rotational Axis

We see first that we can find a neighborhood of the two characteristic singular points that does not contain cycles of a jet approximation $\xi _{\ell }$ of $\hat{\xi }$.

Proposition 5.1

Given $\ell \ge \ell _{\mathcal {M}}+1$, there exist neighborhoods $W_\infty$ of $\gamma _{I_{\infty }^\mathcal {M}}$ and $W_{-\infty }$ of $\gamma _{I_{-\infty }^\mathcal {M}}$ in M such that neither $W_\infty {\setminus } E$ nor $W_{-\infty }{\setminus } E$ contains cycles of $\pi ^*\xi _{\ell }$.

Proof

According to the construction in Sect. 3.3, the point $\gamma _{I_{\infty }^\mathcal {M}}$ is the origin of the chart $(C_{J},(x^{(J)}, y^{(J)}, z^{(J)} ))$ with $J=I_{\infty }^\mathcal {M}$ and $E\cap C_J=\{ z^{(J)}=0 \}$. Being $\mathcal {M}$ an adapted resolution of singularities of $\hat{\xi }$ and by means of Corollary 3.15, we have in a neighborhood of $\gamma _{I_{\infty }^\mathcal {M}}$ that $\xi _\ell ^{(J)}(z^{(J)})=(z^{(J)})^{n^{(J)}} \cdot F(x^{(J)}, y^{(J)}, z^{(J)})$ where $\xi _\ell ^{(J)}=(\pi |_{C_J})^*\xi _\ell$, $n ^{(J)}\in \mathbb {N}_{\ge 1}$ and $F(x^{(J)}, y^{(J)}, z^{(J)})\in \mathbb {R}\{ x^{(J)}, y^{(J)}, z^{(J)}\}$ converges and satisfies $F(0,0,0)\ne 0$. Take a neighborhood $W_\infty$ of $I_{\infty }^\mathcal {M}$ in M where F has a constant sign, positive or negative. We have that the trajectories of $\pi ^*\xi _{\ell }$ in $W_\infty {\setminus } E$ can be parameterized by $z^{(J)}$, which avoids the existence of cycles of $\pi ^*\xi _{\ell }$ in $W_\infty {\setminus } E$. The proof for $\gamma _{I_{-\infty }}^\mathcal {M}$ is analogous. $\square$

5.2 Simple Corner Characteristic Cycles

We prove that cycles of $\pi ^*\xi _{\ell }$ cannot accumulate along corner characteristic cycles. Once again, the argument is to find a function, around such corner characteristic cycle, which is monotonic along the trajectories of $\pi ^*\xi _{\ell }$, if $\ell$ is sufficiently large.

Proposition 5.2

Let $\ell \ge \ell _\mathcal {M}+1$. Consider a corner characteristic cycle $\gamma _I$ of $\hat{\xi }$ in $\mathcal M$. Then, there exists a neighborhood $W_{I}$ of $\gamma _{I}$ in M such that $\pi ^*(\xi _{\ell })$ does not contain cycles in $W_{I}{\setminus } E$.

Proof

By construction, the corner characteristic cycle $\gamma _I$ is given by $\{ z ^{(J)}=\rho ^{(J)} = 0 \}$ for some chart $(C_J,(\theta , z ^{(J)}, \rho ^{(J)})) \in \mathcal {A}$ for which $E\cap C_J=\{\rho ^{(J)}z ^{(J)}=0 \}$. For simplicity, from now on, we remove the super-indices of the coordinates. By definition of $\pi$ being an adapted resolution of singularities, at least one of the two components of the divisor $\{ \rho =0 \}$ and $\{ z=0 \}$ is non-dicritical. More precisely, let $\hat{\eta }_J$ be the two dimensional vector field associated to $\hat{\xi }^{(J)}=(\pi |_{C_J})^*(\hat{\xi })$ and consider $\hat{\eta }_J'=\frac{1}{\rho ^{a} z^{b}} \hat{\eta }_J$ the reduced associated vector field, i.e., $a=n_1^{(J)}, b=n_2^{(J)}$. We have two cases:

• (a) The origin is not a singular point of $\hat{\eta }_J'$ and one of the components, say $\{ z=0 \}$, is the solution of $\hat{\eta }_J'$.

• (b) The origin is a simple singularity of $\hat{\eta }_J'$ and both components are invariant for $\hat{\eta }_J'$.

In the case (a) write

$$\begin{aligned} \hat{\eta }_J'= zF_z(z,\rho )\frac{\partial }{\partial z}+ (\lambda _2 + F_\rho (z,\rho ))\frac{\partial }{\partial \rho }\end{aligned}$$

where $\lambda _2\ne 0$ and $F_z, F_\rho \in \mathbb {R}[\rho ][[z]]\cap \mathbb {R}[z][[\rho ]]$ with $F_\rho (0,0)= 0$. We have that

$$\begin{aligned} \hat{\xi } ^{(J)}(\rho )=\rho ^{a} z^{b}\cdot (\lambda _2 + F_\rho (z,\rho )) \hat{\xi } ^{(J)}(\theta ). \end{aligned}$$

(39)

Since $\ell \ge \ell _\mathcal {M}+1$, Corollary 3.15 implies that:

$$\begin{aligned} \xi _\ell ^{(J)}(\rho ) = \rho ^{a} z^{b}\cdot (\lambda _2 +F_\rho ^\ell (\theta ,z,\rho )) \xi _\ell ^{(J)}(\theta ), \end{aligned}$$

where $F^\ell _\rho$ is analytic and $F^\ell _\rho (\theta ,0,0)=0$. Considering that the monomial $\rho ^{a} z^{b}>0$ for $(z,\rho )\in \mathbb {R}_{>0}^ 2$, and taking into account that $\lambda _2\ne 0$ and $\xi _\ell ^{(J)}(\theta )>0$ along $\gamma _I$, there is a neighborhood $W_I$ of $\gamma _I$ such that $\xi _\ell ^{(J)}(\rho )$ has constant sign in $W_I{\setminus } E$. Hence, the trajectories of $\xi _\ell ^{(J)}$ can be parameterized by $\rho$ in $W_I{\setminus } E$ and thus $\xi _{\ell }^{(J)}$ cannot have cycles in $W_I{\setminus } E$.

In the case (b) being both components of the divisor invariant, we can write:

$$\begin{aligned} \hat{\eta }_J'= ( \lambda _1 z +z F_z(z,\rho ))\frac{\partial }{\partial z}+ (\lambda _2\rho + \rho F_\rho (z,\rho ))\frac{\partial }{\partial \rho }, \end{aligned}$$

where $\lambda _1^ 2+ \lambda _2^2 \ne 0$ and $F_z, F_\rho \in \mathbb {R}[\rho ][[z]]\cap \mathbb {R}[z][[\rho ]]$ satisfy $F_\rho (0,0)=F_z(0,0)= 0$. Suppose without loss of generality that $\lambda _1\ne 0$. Then, we write:

$$\begin{aligned} \hat{\xi } ^{(J)}(z) =\rho ^{a} z^{b+1}\cdot (\lambda _1 + F_z(z,\rho )) \hat{\xi } ^{(J)}(\theta ). \end{aligned}$$

Since $\ell \ge \ell _\mathcal {M}+1$, Corollary 3.15 implies that:

$$\begin{aligned} \xi _\ell ^{(J)}(z)= \rho ^{a} z^{b+1}\cdot (\lambda _1 +F_z^\ell (\theta ,z,\rho )) \xi _\ell ^{(J)}(\theta ), \end{aligned}$$

where $F^\ell _z$ is analytic and $F^\ell _z(\theta ,0,0)=0$. As in the first case, we find that the trajectories of $\xi _\ell ^{(J)}$ can be parameterized by z in $W_I{\setminus } E$ and $\xi _\ell ^{(J)}$ cannot have cycles in $W_I{\setminus } E$. $\square$

5.3 Simple Non-corner Characteristic Cycles

All along this subsection, we suppose that $\gamma _I$ is a non-corner characteristic cycle of $\mathcal {M}$ contained in a chart $C_J$ for which $\{ \rho ^{(J)}=0 \}$ is the equation of $E\cap C_J$ and $\gamma _I=\{ \rho ^{(J)}=0, z^{(J)}=w_{I} \}$ for some $\omega _I\in \mathbb {R}$. Consider the transform $\hat{\xi }^{(J)}=(\pi |_{C_J})^*\hat{\xi }$ in the translated coordinates $(z:=z^{(J)}-w_I,\rho :=\rho ^{(J)})$. Its associated two-dimensional vector field is

$$\begin{aligned} \hat{\eta }_J:= \frac{\hat{\xi }^{(J)}(\rho )}{\hat{\xi }^{(J)}(\theta )}\frac{\partial }{\partial \rho }+ \frac{\hat{\xi }^{(J)}(z)}{\hat{\xi }^{(J)}(\theta )}\frac{\partial }{\partial z}. \end{aligned}$$

More precisely, we write $\hat{\eta }_J=\rho ^a \hat{\eta }_J'$ where $a\ge 0$ and $\hat{\eta }'_J$ is a formal vector field in coordinates $(z,\rho )$ with a simple singularity at the origin, One of the separatrices of $\hat{\eta }'_J$ is the divisor $\{\rho =0\}$ and the other one, denoted by $\hat{\Gamma }_I$, is smooth and transverse to the divisor.

5.3.1 Invariant Formal Surface Along $\gamma _I$

Being $\hat{\Gamma }_I$ a formal non-singular curve transverse to $\{\rho =0\}$, it can be expressed as a formal graph $z={\hat{h}}_I(\rho )$, where ${\hat{h}}_I(\rho )\in \mathbb {R}[[\rho ]]$. Since $L_{\frac{\partial }{\partial \theta }} \hat{\xi }^{(J)}=0$, we have that ${\hat{S}}_I:= \mathbb {S}^1\times \hat{\Gamma }_I$ is a formal invariant non-singular surface of $\hat{\xi }^{(J)}$ supported along the cycle $\gamma _I$. Its vanishing ideal $\text {id}({\hat{S}}_I)$ in the ring $\mathbb {R}[\cos \theta ,\sin \theta ][[z,\rho ]]$ is generated by $H_I(z,\rho ):=z-{\hat{h}}_I(\rho )$. Using this surface, we can also construct a formal invariant surface for the transformed vector field $\xi _{\ell }^{(J)}=(\pi |_{C_J})^*\xi _\ell$ along the characteristic cycle $\gamma _I$, when $\ell$ is sufficiently large. More precisely,

Proposition 5.3

Suppose that $\ell \ge \ell _\mathcal {M}+1$. Then, there is a formal invariant surface ${\hat{S}}_{\ell ,I}$ of $\xi _{\ell }^{(J)}$ along $\gamma _I$ expressed in coordinates $(\theta , z,\rho )$ as the ideal in $\mathbb {R}[\cos \theta , \sin \theta ][[z,\rho ]]$ generated by some series of the form $H_{\ell ,I}(\theta , z,\rho ):=z-h_{\ell ,I}(\theta ,\rho ),$ where $h_{\ell ,I}\in \mathbb {R}[\cos \theta ,\sin \theta ][[\rho ]]$ with $h_{\ell ,I}(\theta ,0)=0.$

Proof

Consider the formal conjugation $\psi _\ell ^*\xi _{\ell }=\hat{\xi }$ (cf. Eq. (3)). From Proposition 3.16, there is a formal automorphism $\psi _\ell ^{(J)}$ defined by:

$$\begin{aligned} (\theta ,z,\rho )\circ \psi _\ell ^{(J)}=(\psi _{\ell }^\theta ,\psi _{\ell }^z,\psi _{\ell }^\rho )=(\theta + O(\rho ^2),z+ O(\rho ^2),\rho + O(\rho ^2)), \end{aligned}$$

conjugating $\hat{\xi }^{(J)}$ to $\xi _{\ell }^{(J)}$ and such that $\psi _\ell ^{(J)}-(\theta ,z,\rho )\in \mathbb {R}[\cos \theta , \sin \theta ,z][[\rho ]]^3$. We consider the formal surface ${\hat{S}}_{\ell ,I}$ whose defining ideal is $\text {id}({\hat{S}}_{\ell ,I})=({\widetilde{H}}_{\ell ,I}(\theta , z,\rho ))$ where

$$\begin{aligned} {\widetilde{H}}_{\ell ,I}(\theta , z,\rho ):=(\psi _\ell ^{(J)})^*( H_I)= H_I\circ \psi _\ell ^{(J)} =\psi _{\ell }^z-{\hat{h}}_{I}(\psi _{\ell }^\rho )\in \mathbb {R}[\cos \theta , \sin \theta ,z][[\rho ]]. \end{aligned}$$

Using that $\frac{\partial {\widetilde{H}}_{\ell ,I}}{\partial z}(0,0,0)\ne 0$ and applying the implicit function theorem to ${\widetilde{H}}_{\ell ,I}$, we find an expression of the form $H_{\ell ,I}=z-h_{\ell ,I}(\theta ,\rho )$ for a generator of $id({\hat{S}}_{\ell ,I})$, with $h_{\ell ,I}\in \mathbb {R}[\cos \theta , \sin \theta ][[\rho ]]$. $\square$

5.3.2 Poincaré First-Return Map Associated to $\gamma _I$

By Remark 3.14, $\gamma _I$ is a trajectory of the vector field $\xi _\ell ^{(J)}=(\pi |_{C_J})^*\xi _\ell$ for $\ell \ge \ell _{\mathcal { M}}+1$. Let $P=P_{\ell ,I}:\Delta \rightarrow \{ \theta =0 \}$ be the Poincaré first-return map of $\xi ^{(J)}_\ell$ relatively to $\gamma _I$, where $\Delta$ is a sufficiently small neighborhood of $(z,\rho )=(0,0)$ in $\{ \theta =0 \}$ in which P is analytic.

Notice that the Poincaré map does not depend on the parametrization of the trajectories of the vector field, and hence, we can define it using any equivalent vector field. In particular, we are going to consider the vector field ${\tilde{\xi }}_\ell ^{(J)}$ equivalent to $\xi _\ell ^{(J)}$ obtained by the multiplication by the inverse of $\xi _\ell ^{(J)}(\theta )$. That is, we put

$$\begin{aligned} {\tilde{\xi }}_\ell ^{(J)}=\frac{\partial }{\partial \theta }+ \chi ,\quad \text {where } \ \chi =\frac{\xi _\ell ^{(J)}(z)}{\xi _\ell ^{(J)}(\theta )} \frac{\partial }{\partial z}+ \frac{\xi _\ell ^{(J)}(\rho )}{\xi _\ell ^{(J)}(\theta )}\frac{\partial }{\partial \rho }. \end{aligned}$$

(40)

Notice that the components of $\chi$ are the right members of the system of ODEs $\eta _{\ell , J}$ introduced in Sect. 3.5. They belong to the $\mathbb {R}-$algebra $\mathbb {R}[\cos \theta , \sin \theta ]\{z,\rho \}$ (by Remark 3.14). Thus, we consider ${{\widetilde{\xi }}}_\ell ^{(J)}$ as an analytic vector field on the domain $(\theta ,z,\rho )\in \mathbb {R}\times (-\delta ,\delta )^2$, for some $\delta >0$, $2\pi -$periodic in the variable $\theta$. Moreover, from Corollary 3.15, we have that $\rho$ divides $\chi$ and hence ${{\widetilde{\xi }}}^{(J)}_\ell |_{E}=\frac{\partial }{\partial \theta }$.

Denote by $\Phi ^t:=\Phi ^t_{{\tilde{\xi }}_{\ell }^{(J)}}$ the flow map of ${\tilde{\xi }}_{\ell }^{(J)}$. It is defined and analytic for $(t,(\theta ,z,\rho ))\in (-\varepsilon ,2\pi +\varepsilon )\times \left( (-\varepsilon ,2\pi +\varepsilon ) \times V\right)$ where V is a neighborhood of $0\in \mathbb {R}^2$. Using that ${{\widetilde{\xi }}}_\ell ^{(J)}(\theta )=1$, we obtain

$$\begin{aligned} \Phi ^t(\theta ,z,\rho )= (\theta +t, \Psi ^t_z(\theta ,z,\rho ),\Psi ^t_\rho (\theta ,z,\rho )), \end{aligned}$$

(41)

that is, the angle $\theta$ is the natural time for ${\tilde{\xi }}_{\ell }^{(J)}$. By definition, the Poincaré map is given by

$$\begin{aligned} P(z,\rho )=(\Psi _z^{2\pi }(0,z,\rho ),\Psi _\rho ^{2\pi }(0,z,\rho )). \end{aligned}$$

(42)

We are going to express the flow via the exponential map. To be precise, given any $G\in \mathbb {R}[\cos \theta , \sin \theta ][[z,\rho ]]$, we define:

$$\begin{aligned} \text {Exp}(t{\tilde{\xi }}_{\ell }^{(J)})(G):=\displaystyle \sum _{i=0}^{\infty } \frac{t^i}{i!} ({\tilde{\xi }}_{\ell }^{(J)})^{(i)} (G), \end{aligned}$$

where, for any vector field $\zeta$, $\zeta ^{(0)}(G)=G$ and $\zeta ^{(i)}(G)=\zeta (\zeta ^{(i-1)}(G))$, if $i\ge 1$. Taking into account the above properties of the components of ${\tilde{\xi }}_{\ell }^{(J)}$, it is immediate to check that $\text {Exp}(t{\tilde{\xi }}_{\ell }^{(J)})(G)\in \mathbb {R}[\cos \theta , \sin \theta ][[t,z,\rho ]]$. In the following result, we get some useful properties of this exponential map and its relation with the flow map. Notice first that, if $G\in \mathbb {R}[\cos \theta , \sin \theta ][[z,\rho ]]$, then the composition $G\circ \Phi ^t$, due to the analyticity of $\Phi ^t$, has a formal Taylor expansion at $t=0$, denoted by $T_0(G\circ \Phi ^t)$, a formal power series in variables $(t,z,\rho )$, with analytic functions of $\theta \in (-\varepsilon , 2\pi +\varepsilon )$ as coefficients.

Proposition 5.4

Let $G\in \mathbb {R}[\cos \theta , \sin \theta ][[z,\rho ]]$. We have:

• (1) $T_0(G\circ \Phi ^t)={\text {Exp}}(t{\tilde{\xi }}_{\ell }^{(J)})(G) \in \mathbb {R}[\cos \theta , \sin \theta ][[t,z,\rho ]]$

• (2) For any $t_0\in [0,2\pi ]$, the expression ${\text {Exp}}(t_0{\tilde{\xi }}_{\ell }^{(J)})(G)= \displaystyle \sum _{i=0}^{\infty } \frac{t^i_0}{i!} ({\tilde{\xi }}_{\ell }^{(J)})^{(i)} (G)$ has a sense as a series in $\mathbb {R}[\cos \theta , \sin \theta ][[z,\rho ]]$ and we have

  $$\begin{aligned} G\circ \Phi ^{t_0}={\text {Exp}}(t_0{\tilde{\xi }}_{\ell }^{(J)})(G) \end{aligned}$$

  (43)

Proof

We prove (1) with the same arguments as the case in Loray’s text for holomorphic vector fields [23, p. 15]: expand $G{\circ }\Phi ^t$ as a Taylor series in t at $t=0$, so that we get

$$\begin{aligned} T_{0}(G{\circ }\Phi ^{t})=\displaystyle \sum _{i=0}^\infty \frac{t^i}{i!} \frac{\partial ^i(G{\circ } \Phi ^{t})}{\partial t^i}\Bigg \vert _{t=0}, \end{aligned}$$

and check that $\frac{\partial ^i(G{\circ } \Phi ^t)}{\partial t^i}= ({\tilde{\xi }}_{\ell }^{(J)})^{(i)} (G)\circ \Phi ^t$ for any $i\ge 1$.

Let us prove item (2). First, we show that there exists $\alpha >0$ such that (2) is true for any $t_0\in [0,\alpha ]$. For that, consider the particular case where G is either the coordinate z or $\rho$ (with the notations of (41), $z\circ \Phi ^{t_0}=\Psi _z^{t_0}$ and $\rho \circ \Phi ^{t_0}=\Psi _\rho ^{t_0}$). By analyticity of these functions and by item (1), we get that $\text {Exp}( t {\tilde{\xi }}_\ell ^{(J)})(z), \text { Exp}(t{\tilde{\xi }}_\ell ^{(J)})(\rho )\in \mathbb {R}[\cos \theta , \sin \theta ]\{ t,z,\rho \}$. More precisely, they belong to $\mathbb {R}[\cos \theta , \sin \theta ]\{ t\}_\beta [[z,\rho ]]$ for some $\beta >0$ (recall the notations stated in Sect. 1, that is, all coefficients in $\mathbb {R}[\cos \theta , \sin \theta ]\{ t\}$ have a common radius of convergence). We conclude that $\Psi _z^{t_0}=z\circ \Phi ^{t_0}= \text {Exp}(t_0{\tilde{\xi }}_{\ell }^{(J)})(z)$ and $\Psi _\rho ^{t_0}=\rho \circ \Phi ^{t_0}= \text {Exp}(t_0{\tilde{\xi }}_{\ell }^{(J)})(\rho )$ for any $t_0\in [0,\alpha ]$ with $0<\alpha <\beta$.

Let $G\in \mathbb {R}[\cos \theta , \sin \theta ][[z,\rho ]]$ be any formal series and write

$$\begin{aligned} G=\displaystyle \sum _{u,v} G_{uv}(\theta )z^u\rho ^v, \text { with } G_{uv}(\theta )\in \mathbb {R}[\cos \theta , \sin \theta ]. \end{aligned}$$

Consider the series

$$\begin{aligned} {\bar{G}}= \displaystyle \sum _{u,v} G_{uv}(\theta +t)z^u\rho ^v \end{aligned}$$

which belongs to $\mathbb {R}[\cos \theta , \sin \theta ]\{ t \}_\beta [[ z,\rho ]]$ since each $G_{uv}(\theta )$ is a trigonometric polynomial. Taking into account the expression of the flow $\Phi ^t$, we have that $G\circ \Phi ^t$ is the result of substituting in the series $\bar{G}$ the variables $z,\rho$ by $\Psi ^t_z,\Psi ^t_\rho$, respectively. Since the series $\Psi _z^t,\Psi _\rho ^t$ belong to $\mathbb {R}[\cos \theta , \sin \theta ]\{ t \}_\beta [[ z,\rho ]]$ and have positive order with respect the variables $z,\rho$, substitution has perfect sense and provides an element in $\mathbb {R}[\cos \theta , \sin \theta ]\{ t \}_\beta [[ z,\rho ]]$. Since, by item (1), $T_0(G\circ \Phi ^t)$ coincides with $\text {Exp}(t\widetilde{\xi }_\ell ^{(J)})(G)$ as a series in $\mathbb {R}[\cos \theta , \sin \theta ][[t,z,p]]$, we conclude item (2) and expression (43) for $t_0\in [0,\alpha ]$. Notice that we can choose $\alpha >0$ which does not depend on G. Let us show that we can extend the property(43) to any $t_0\in [0,2\alpha ]$ (and hence similar extensions will prove (2)). Let $t_0\in [\alpha ,2\alpha ]$ and write $t_0=s_0+\alpha$, where $s_0\in [0,\alpha ]$. We have $G\circ \Phi ^{t_0}=(G\circ \Phi ^{s_0})\circ \Phi ^\alpha$. Applying (43) for the values $s_0$ and $\alpha$, and for G and $G\circ \Phi ^{s_0}$, respectively, we get

$$\begin{aligned} \begin{aligned} G\circ \Phi ^{t_0}&=\displaystyle \sum _i \frac{\alpha ^i}{i!} ({\tilde{\xi }}_{\ell }^{(J)})^{(i)} (G\circ \Phi ^{s_0})= \sum _i \frac{\alpha ^i}{i!} ({\tilde{\xi }}_{\ell }^{(J)})^{(i)} \left( \sum _j \frac{s_0^j}{j!} ({\tilde{\xi }}_{\ell }^{(J)})^{(j)} (G) \right) \\&= \sum _k \left( \sum _{i+j=k} \frac{\alpha ^i}{i!} \frac{s_0^j}{j!}({\tilde{\xi }}_{\ell }^{(J)})^{(k)}(G) \right) \\ &= \sum _k \frac{(\alpha +s_0)^k}{k!}({\tilde{\xi }}_{\ell }^{(J)})^{(k)}(G) = \text {Exp}(t_0 {\tilde{\xi }}_{\ell }^{(J)})(G), \end{aligned} \end{aligned}$$

as it was to be proved. $\square$

We can now prove two important features of the Poincaré map.

Lemma 5.5

There exists $\ell _I$ such that, for any $\ell \ge \ell _I$, the Poincaré map $P=P_{\ell ,I}$ satisfies:

• (a) P is tangent to the identity but $P\ne Id$ as a germ of diffeomorphisms at $(0,0)\in \Delta$.

• (b) The formal curve $\Gamma =\Gamma _{\ell ,I}:=\hat{S}_{\ell ,I}\cap \Delta$ is invariant for P.

Proof

Recall that the two-dimensional formal vector field $\hat{\eta }_J$ associated to the formal vector field $\hat{\xi }^{(J)}$ has an adapted simple singularity corresponding to the characteristic cycle $\gamma _I$. As mentioned, the defining ideal of the formal curve $\hat{\Gamma }_I$ is generated by $z-{\hat{h}}_I(\rho )$, where ${\hat{h}}_I(\rho )=\sum _{i\ge 1}a_i\rho ^i$.

Therefore, we can write $\hat{\eta }_J=\rho ^n\hat{\eta }'_J$ with

$$\begin{aligned} \hat{\eta }_J'= (z-{\hat{h}}_{I}(\rho ))^r \left( (\lambda _1 (z-a_1\rho )+ B_1(z,\rho )) \frac{\partial }{\partial z}+ (\lambda _2\rho + B_2(z,\rho ))\frac{\partial }{\partial \rho }\right) , \end{aligned}$$

where $n=n_1^{(J)}$, $r\in \mathbb {N}_{\ge 0}$, $(\lambda _1,\lambda _2)\ne (0,0)$ and $B_i\in \mathbb {R}[z][[\rho ]]$ has order greater or equal than 2 for $i=1,2$. Up to making a new admissible blowing-up with center $\gamma _I$, we may assume that $\rho$ divides $B_1$ and that $\rho ^2$ divides $B_2$.

Define $\ell _I=\ell _{\mathcal { M}}+r+1$. Applying Corollary 3.15 to $k=r+1$, we get, for any $\ell \ge \ell _{\mathcal { M}}+k=\ell _I$:

$$\begin{aligned} j_{r+1}^\rho (\eta _{\ell , J}')=j_{r+1}^\rho (\hat{\eta }_{ J}'). \end{aligned}$$

(44)

Assume first that $\lambda _1\ne 0$. From (44), and taking into account that $\chi$ coincides with the two dimensional system $\eta _{\ell ,J}$ (cf. Eq. (40)), we obtain

$$\begin{aligned} j_{n}^{\rho }( {\tilde{\xi }}_\ell ^{(J)}) =\frac{\partial }{\partial \theta }+ \lambda _1\rho ^n z^{r+1} \frac{\partial }{\partial z}\end{aligned}$$

(45)

Using this in the computation of the exponential $\text {Exp}(t {\tilde{\xi }}_{\ell }^{(J)})(z)$, written as a series as

$$\begin{aligned} \text {Exp}(t {\tilde{\xi }}_{\ell }^{(J)})(z)=z+tQ_1+ t^2Q_2+ \cdots , \quad \text {with } Q_j\in \mathbb {R}[\cos \theta , \sin \theta ][[z,\rho ]], \end{aligned}$$

one can show by recurrence that

$$\begin{aligned} \begin{aligned} Q_1&=\rho ^n (\lambda _1 z^{r+1}+O(\rho )), \\ Q_j&= O(\rho ^{jn}), \ j\ge 2. \end{aligned} \end{aligned}$$

Using Proposition 5.4 and Eq. (42), we deduce, since $\lambda _1\ne 0$ and $n\ge 1$,

$$\begin{aligned} z\circ P(z,\rho ) = \Psi _z^{2\pi } (0,z,\rho )= \text {Exp}(2\pi {\tilde{\xi }}_{\ell }^{(J)})(z)= z+\rho ^n(2\pi \lambda _1 z^{r+1} + O(\rho ))\ne z. \end{aligned}$$

This proves (a) if $\lambda _1\ne 0$.

On the contrary, if $\lambda _1= 0$ and $\lambda _2\ne 0$, we obtain

$$\begin{aligned} { j_{n+1}^\rho ({\tilde{\xi }}^{(J)}_\ell )}=\frac{\partial }{\partial \theta }+ \rho ^{n+1}g(z)\frac{\partial }{\partial z}+ \lambda _2 \rho ^{n+1} z^{r} \frac{\partial }{\partial \rho }, \end{aligned}$$

where $g(z)\in \mathbb {R}\{ z \}$. We deduce, similarly, that, if we write again

$$\begin{aligned} \text {Exp}(t {\tilde{\xi }}_{\ell }^{(J)})(\rho )=\rho +tQ_1+ t^2Q_2+ \cdots , \quad \text {with } Q_j\in \mathbb {R}[\cos \theta , \sin \theta ][[z,\rho ]], \end{aligned}$$

then $Q_1=\rho ^{n+1}(\lambda _2 z^r + O(\rho ))$ and $Q_j=O(\rho ^{nj+1})$ if $j\ge 2$. Hence,

$$\begin{aligned} \rho \circ P(z,\rho ) = \Psi _\rho ^{2\pi } (0,z,\rho )= \text {Exp}(2\pi {\tilde{\xi }}_{\ell }^{(J)})(\rho )= \rho +\rho ^{n+1}(2\pi \lambda _2 z^{r} + O(\rho ))\ne \rho , \end{aligned}$$

and (a) equally holds.

Let us show (b). Let $H=H_{\ell ,I}\in \mathbb {R}[\cos \theta , \sin \theta ][[z,\rho ]]$ be a generator of the ideal of the invariant surface ${\hat{S}}_{\ell ,I}$ obtained in Proposition 5.3. The series $g(z,\rho ):=H(0,z,\rho )$ is a generator of the formal plane curve $\Gamma =\hat{S}_{\ell ,I}\cap \{ \theta =0 \}$. We need to check that the composition $g\circ P$ is divisible by g. Using Proposition 5.4, we have

$$\begin{aligned} H\circ \Phi ^{2\pi } = \displaystyle \sum _{i\ge 0} \frac{(2\pi )^i}{i!} ({\tilde{\xi }}_{\ell }^{(J)}) ^{(i)}(H). \end{aligned}$$

(46)

Since $\hat{S}_{\ell ,I}$ is invariant for ${\tilde{\xi }}_{\ell }^{(J)},$ we have ${\tilde{\xi }}_{\ell }^{(J)}(H)\in \text {id}(\hat{S}_{\ell ,I})$, that is, H divides ${\tilde{\xi }}_{\ell }^{(J)}(H)$. By recurrence, H divides $({\tilde{\xi }}_{\ell }^{(J)}) ^{(i)}(H)$ for any $i\ge 0$. Thus, from Eq. (46), we get

$$\begin{aligned} H\circ \Phi ^{2\pi }=H\cdot K, \ \text {where }K\in \mathbb {R}[\cos \theta , \sin \theta ][[z,\rho ]]. \end{aligned}$$

We conclude, using that $P(z,\rho )=\Phi ^{2\pi }(0,z,\rho )$,

$$\begin{aligned} g\circ P=(H\circ \Phi ^{2\pi })|_{ \{ \theta =0 \} }= H(0,z,\rho ) K(0,z,\rho ) = g \cdot {\tilde{K}},\ {\tilde{K}}\in \mathbb {R}[[z,\rho ]], \end{aligned}$$

as we wanted to prove. $\square$

5.3.3 Periodic Orbits of the Poincaré Map Around the Invariant Curve

We are interested in the periodic orbits of P near (0, 0) since, as we know, they correspond to cycles of $\xi _\ell ^{(J)}$ near $\gamma _I$. As it was the case for defining cycles, periodic orbits depend on the domain of (a representative of) P. To be more precise, if P is defined in some neighborhood W of (0, 0), a periodic orbit in W is a finite set $\{p_i\}_{i=0}^{n-1}$ contained in W such that $p_i=P(p_{i-1})$ for $i=1,\ldots , n-1$ and $p_0=P(p_{n-1})$.

Denote by $\text {Fix}(P)$ the (germ of the) locus of fixed points of P. It is an analytic set with empty interior (since $P\ne id$), thus either reduced to the origin or an analytic curve with finitely many branches. We can distinguish two different situations:

• (a) The invariant curve $\Gamma$ is not contained in $\text {Fix}(P)$.

• (b) The invariant curve $\Gamma$ is contained in $\text {Fix}(P)$.

In particular, in case (b), $\Gamma$ is a real branch of $\text {Fix}(P)$, and thus it converges. In both cases, we investigate periodic orbits of P in some “neighborhood” of $\Gamma$. To be precise, consider a parametrization of $\Gamma$ of the form $z=h(\rho )$, with $h(\rho )\in \mathbb {R}[[\rho ]]$ and $h(0)=0$. A conic neighborhood of $\Gamma$ is a set of the form

$$\begin{aligned} \Sigma _{N,\delta } (\Gamma )= \Sigma _{ N,\delta }^{(z,\rho )}(\Gamma ):=\{ (z,\rho ): |z-j_N(h(\rho ))|<\rho ^N, 0<\rho <\delta \}. \end{aligned}$$

where $N\in \mathbb {N}_{\ge 1}$ and any $\delta >0$ sufficiently small.

Remark 5.6

By its definition, these conic neighborhoods depend on the chosen coordinates. However, after a simple change of variables consisting in a transformation of type ${\bar{z}}=z +\alpha (\rho )$ with $\alpha (\rho )\in \mathbb {R}[[\rho ]]$, the sets $\Sigma ^{(z,\rho )}_{N,\delta }$ and $\Sigma ^{({\bar{z}},\bar{\rho })}_{N,\delta }$ coincide exactly under this change.

Case (a) In this case, we prove that there are no periodic orbits in a conic neighborhood of $\Gamma$. The arguments are inspired by the papers [21, 22], devoted to treat this case for holomorphic diffeomorphisms.

Lemma 5.7

Suppose that $\Gamma$ is not contained in ${\text {Fix}}(P)$. Then, there is some $N\in \mathbb {N}_{\ge 1}$ and some $\delta >0$ such that (a representative of) P does not have periodic orbits in $\Sigma _{ N,\delta }(\Gamma )$.

Proof

First, since the divisor is contained in $\text {Fix}(P),$ we have that $\rho \circ P -\rho$ can be divided by $\rho$. On the other hand, being $\Gamma$ invariant for P, there is a formal diffeomorphism $\Theta (\rho )=\rho + O(\rho ^2)\in \mathbb {R}[[\rho ]]$ satisfying:

$$\begin{aligned} P(h(\rho ), \rho )=(h(\Theta (\rho )), \Theta (\rho )). \end{aligned}$$

The formal diffeomorphism $\Theta$ is called the restriction of P to $\Gamma$, denoted by $P|_{\Gamma }:=\Theta$ (see [21]). The order of $P|_\Gamma$, defined as $\text {ord}_\rho (\Theta (\rho )-\rho )-1$, does not depend on the coordinates nor the parametrization $(h(\rho ),\rho )$ of $\Gamma$. In this case, it is a natural number $m<\infty$, because otherwise, $\Theta (\rho )=\rho$ and $\Gamma$ would be contained in $\text {Fix}(P)$. We deduce that there is a maximal $k\ge 1$ such that $\rho ^k$ divides $\rho \circ P-\rho$, so that we can write

$$\begin{aligned} \rho \circ P-\rho =\rho ^k(A(\rho )+ zB(z,\rho )), \end{aligned}$$

(47)

where $A\in \mathbb {R} \{ \rho \}$, $B\in \mathbb {R} \{z, \rho \}$ and $\rho ^k(A(\rho )+ zB(z,\rho ))\ne 0$. Up to taking new coordinates $({\bar{z}}, \rho )$ with ${\bar{z}} =z-j_{m+1}(h(\rho ))$ (which we rename $(z,\rho )$ for simplicity) and using Remark 5.6, we may assume from the beginning that $\text {ord}_\rho (h(\rho ))\ge m+2$. From equation (47), the definition of $\Theta =P|_\Gamma$ and its order m, we have

$$\begin{aligned} \alpha \rho ^{m+1} + \cdots = \Theta (\rho )-\rho =\rho ^k (A(\rho )+ h(\rho )B(h(\rho ),\rho ) ), \ \alpha \ne 0, \end{aligned}$$

which implies that $A(\rho )=\rho ^s {{\widetilde{A}}}(\rho )$ with $k+s=m+1$ and ${\widetilde{A}}(0)=\alpha$ (in particular s is finite). Put $N=m+1$ and let us prove the required property for a cone $\Sigma _{ N,\delta }=\Sigma _{ N,\delta }^{(z,\rho )}(\Gamma )$ with some $\delta >0$. Notice first that, for the chosen coordinates $(z,\rho )$, we have $j_N(h(\rho ))=0$, so $\Sigma _{ N,\delta }$ is given simply by equations $|z|<\rho ^N$ and $0<\rho <\delta$. On the other hand, $N=k+s>s$, since $k>0$. Assume for instance that $\alpha <0$ (analogous arguments apply if $\alpha >0$). Take a preliminary $\delta _1>0$ such that ${{\widetilde{A}}}(\rho )<\frac{\alpha }{2}$ if $\rho <\delta _1$ and let $K>0$ be a bound for |B| in a neighborhood of (0, 0) that contains $\Sigma _{N,\delta _1}$. We have in $\Sigma _{N,\delta _1}$

$$\begin{aligned} \rho ^s {{\widetilde{A}}}(\rho )+ zB<\rho ^s{{\widetilde{A}}}(\rho )+\rho ^N K <\rho ^s(\frac{\alpha }{2}+\rho ^{N-s}K). \end{aligned}$$

(48)

Now take $\delta <\delta _1$ such that $\delta ^{N-s}<\frac{|\alpha |}{4K}$, and hence we obtain from (48)

$$\begin{aligned} \rho \circ P-\rho =\rho ^k (\rho ^s {{\widetilde{A}}}(\rho )+ zB)<\rho ^k (\rho ^s\frac{\alpha }{4})<0 \ \text {in }\Sigma _{ N,\delta }. \end{aligned}$$

(49)

We conclude that, if $\{ p_0, p_1=P(p_0), p_2=P(p_1), \ldots \}$ is an orbit (finite or not) of P completely contained in $\Sigma _{ N,\delta }$, then the sequence of $\rho -$coordinates decreases strictly:

$$\begin{aligned} \rho (p_0)>\rho (p_1)>\cdots \end{aligned}$$

This proves the result. $\square$

Case (b) Suppose that $\Gamma$ is convergent and contained in $\text {Fix}(P)$. Convergence means that $\Gamma$ is an analytic curve given by a graph $\Gamma = \{ (h(\rho ),\rho ): \rho \in [0,\varepsilon ) \}$, where $h(\rho )\in \mathbb {R}\{ \rho \}$. Up to taking new analytic coordinates $(z-h(\rho ),\rho )$, we will assume that $h(\rho )\equiv 0$. Thus, the conic neighborhoods $\Sigma _{ N,\delta }=\Sigma _{ N,\delta }^{(z,\rho )}(\Gamma )$ will be simply defined by the equations $|z|<\rho ^N$ and $0<\rho <\delta$. With these assumptions, we prove the following result.

Lemma 5.8

Suppose that $\Gamma = \{ z=0 \}\subset {\text {Fix}}(P)$. Then, there is $N\in \mathbb N_{\ge 1}$ and $\delta >0$ such that the fixed points of the set $\Gamma \cap \Sigma _{ N,\delta }$ are the only periodic orbits of P in $\Sigma _{ N,\delta }$.

Proof

The two coordinate axis $\{ \rho =0\}$ and $\{ z= 0\}$ are contained in $\text {Fix}(P)$. Thus, both components $(z\circ P- z,\rho \circ P -\rho )$ of the map $P-Id$ are divisible by a positive power of z and by a positive power of $\rho$. In particular, we can write $z\circ P= z(1+ \psi (z,\rho ))$ where $\rho$ divides $\psi$. From this, we prove the following observation.

Claim

There is a neighborhood V of (0, 0) such that if $\{ p_0,p_1,\ldots \}$ is an orbit of P contained in V, then the sign of the z coordinate of its elements is constant.

Proof of the claim:

Since $\psi (0,0)=0$, we can consider a neighborhood V where we have $(1+\psi (z,\rho ))>\frac{1}{2}$. Hence, $\text {Sign}(z\circ P)=\text {Sign}(z(1+\psi (z,\rho )))=\text {Sign}(z)$ and the claim follows.

On the other hand, since $P\ne Id$ (cf. Lemma 5.5), the two components $z\circ P-z, \rho \circ P-\rho$ cannot be identically zero simultaneously. Suppose that $\rho \circ P-\rho \ne 0$. Then we can write

$$\begin{aligned} \rho \circ P -\rho = \rho ^{k_1}z^{k_2}(A(\rho )+ zB(z,\rho )), \end{aligned}$$

(50)

where $k_1,k_2\in \mathbb {N}_{\ge 1}$ and $A(\rho )$ is a convergent non-zero series. We write $A(\rho )=\rho ^s (\alpha + \cdots ),$ where $s\ge 0$ and $\alpha \ne 0$. Analogously as in the proof of Lemma 5.7, if N is any given natural number with $N>s$, then there exists $\delta >0$ such that the function $A(\rho )+ zB(z,\rho )$ has constant non-zero sign on $\Sigma _{ N,\delta }$. Taking into account the claim above, if $\delta$ is sufficiently small so that $\Sigma _{ N,\delta }\subset V$, we conclude form Eq. (50) that if $\{ p_0,p_1,\ldots \}$ is an orbit of P contained in $\Sigma _{ N,\delta }{\setminus } \Gamma = \left( \Sigma _{ N,\delta }\cap \{ z>0\} \right) \cup \left( \Sigma _{ N,\delta }\cap \{ z<0\} \right)$, then the sequence $\{ \rho (p_0), \rho (p_1), \ldots \}$ is strictly increasing or strictly decreasing. This proves the lemma in case $\rho \circ P -\rho \ne 0$. On the contrary, if $\rho \circ P -\rho = 0$ but $z\circ P - z\ne 0$, we obtain analogously that the $z-$coordinate of elements of an orbit in $\Sigma _{ N,\delta }{\setminus } \Gamma$ is strictly increasing or strictly decreasing, which proves the lemma equally. $\square$

6 Proof of the Main Theorem

In this section we provide a proof of Theorem 1.1. It is enough to prove the result for some jet approximation $\xi _{\ell }$ of a formal normal form $\hat{\xi }$ of $\xi$, since all those vector fields are locally analytically conjugated to $\xi$ at $0\in \mathbb {R}^3$.

Fix an adapted reduction of singularities $\mathcal { M}=(M,\pi ,\mathcal {A}, \mathcal {D})$ of $\hat{\xi }$ given by Proposition 3.10 and denote by $\mathcal {D}^{nc}$ the subset of $\mathcal {D}$ consisting on non-corner characteristic cycles. For any $\gamma _I\in \mathcal {D}^{nc}$, we consider a chart $C_J$ and coordinates $(\theta , {z}^{(I)}:=z^{(J)}-\omega _I, \rho ^{(I)}=\rho ^{(J)})$ as in Sect. 5.3, such that $\gamma _I\subset C_J$ and given by $\gamma _I=\{ z^{(I)}=0, \rho ^{(I)}=0 \}$. Let $\hat{\Gamma }_I$ be the formal plane curve at the origin $(z^{(I)},\rho ^{(I)})=(0,0)$ of $\{ \theta =0 \}$, invariant for the associated vector field $\hat{\eta }_J$ and transversal to the divisor $\{ \rho ^{(I)}=0 \}$. Consider the parametrization of $\hat{\Gamma }_I$ given in these coordinates as $z={\hat{h}}_I(\rho ^{(I)})$, ${\hat{h}}_I\in \mathbb {R}[[\rho ^{(I)}]]$, ${\hat{h}}_I(0)=0$. Consider also the formal surface ${\hat{S}}_I=\mathbb {S}^1\times \hat{\Gamma }_I$ invariant for $\hat{\xi } ^{(J)}$ and given by the same equation $z={\hat{h}}_I(\rho ^{(I)})$, but considering it in $\mathbb {R}[\cos \theta , \sin \theta ,z][[\rho ]]$. Let $\ell$ be the first natural number such that $\ell \ge \ell _I$ for any $\gamma _I \in \mathcal {D}^{nc}$, where $\ell _I$ is given in Lemma 5.5 (notice that $\ell \ge \ell _{\mathcal { M}}+1$). Denote by ${\hat{S}}_{\ell ,I}$ the formal invariant surface of $\xi _\ell ^{(J)}$ given in Proposition 5.3 with defining ideal $id({\hat{S}}_{\ell ,I})=(H_{\ell ,I})$, where $H_{\ell ,I}=z^{(I)}-h_{\ell ,I}(\theta , \rho ^{(I)})$ with $h_{\ell ,I}\in \mathbb {R}[\cos \theta , \sin \theta ][[\rho ^{(I)}]]$. Denote by $P_{\ell ,I}$ the Poincaré map of $\xi _\ell ^{(J)}$ along $\gamma _I$ defined in some neighborhood of the origin of $\{ \theta =0 \}$ and $\Gamma _{\ell ,I}={\hat{S}}_{\ell ,I}\cap \{ \theta =0 \}$ its corresponding formal invariant curve (by Lemma 5.5). Denote furthermore $\mathcal {D}^{nc}=\mathcal {D}^{nc,nfix}\cup \mathcal {D}^{nc,fix}$ as a disjoint union, where $\gamma _I\in \mathcal {D}^{nc,fix}$ if and only if $\Gamma _{\ell ,I}\subset \text {Fix}(P_{\ell ,I})$.

Apply Lemmas 5.7 or 5.8 according to whether $\gamma _I\in \mathcal {D}^{nc,nfix}$ or $\gamma _I\in \mathcal {D}^{nc,fix}$, and get conic neighborhoods $\Sigma _{N_{\ell ,I},\delta }=\Sigma _{N_{\ell ,I},\delta }(\Gamma _{\ell ,I})$ (in coordinates $(z^{(I)},\rho ^{(I)})$ and where we have unified $\delta >0$) where $P_{\ell ,I}$ has no periodic orbits except for the fixed points $\Gamma _{\ell ,I}\cap \Sigma _{N_{\ell ,I},\delta }$ when $\gamma _I\in \mathcal {D}^{nc,fix}$.

This information is of course relevant in order to describe the union of cycles of the transform $\xi _{\ell }^{(J)}$ near $\gamma _I$, but it is not enough a priori, since $\Sigma _{N_{\ell ,I},\delta }$ is not a full neighborhood of the origin at $\{ \theta =0 \}$. By further blowing-ups along the characteristic cycles $\gamma _I$, one can eventually open these conic neighborhoods to full neighborhoods of $\gamma _I$ but, if the order of tangency $N_{\ell ,I}$ is too large, the sequence of blowing-ups to be done may not follow the formal curve $\hat{\Gamma }_I$, and thus we could skip the context of sequences of admissible blowing-ups. We can take a bigger $\ell '$ so that the formal curve $\Gamma _{\ell ',I}$ approximates $\hat{\Gamma }_I$ better than the curve $\Gamma _{\ell ,I}$ does. But the order $N_{\ell ',I}$ may increase with $\ell '$ a priori.

In the strategy that follows we overcome these difficulties. We have to consider first the same kind of conic three-dimensional neighborhoods of the surfaces ${\hat{S}}_{\ell ,I}$. With more generality, consider coordinates $(\theta ,z,\rho )$ in $\mathbb {S}^1\times \mathbb {R}\times \mathbb {R}_{\ge 0}$, put $\gamma =\mathbb {S}^1\times \{ (0,0) \}$ and let S be a formal non-singular surface along $\gamma$ given by an equation of the form

$$\begin{aligned} z-h(\theta ,\rho )=0, \ \text {where } h(\theta ,\rho )\in \mathbb {R}[\cos \theta , \sin \theta ][[\rho ]]. \end{aligned}$$

For $N\in \mathbb {N}_{\ge 0}$ and constants $C,\delta >0$, we define

$$\begin{aligned} {{\widetilde{\Sigma }}}_{N,\delta ,C} (S)= {{\widetilde{\Sigma }}}_{N,\delta ,C}^{(\theta ,z,\rho )} (S)= \{ (\theta ,z,\rho ) \, \ \ |z-j_N^\rho (h(\theta ,\rho )) |< C \rho ^N, 0<\rho <\delta \}. \end{aligned}$$

In particular, notice that $\Sigma _{N,\delta }(\Gamma _{\ell ,I})= {{\widetilde{\Sigma }}}_{N,\delta ,1}({\hat{S}}_{\ell ,I})\cap \{ \theta =0 \}$. We need two results, the first one is just a remark that follows from the construction of sequences of admissible blowing-ups in Sect. 3 and the definition of adapted simple singularities.

Remark 6.1

If $\gamma _I\in \mathcal {D}^{nc}$ and $\widetilde{\mathcal {M}}=({\widetilde{M}},{\widetilde{\pi }},\widetilde{\mathcal {A}},\widetilde{\mathcal {D}})$ is the sequence of admissible blowing-ups obtained from $\mathcal {M}$ by blowing-up along $\gamma _I$ (that is, ${\widetilde{\pi }}=\pi \circ \sigma _{\gamma _{I}}$), then $\widetilde{\mathcal {M}}$ is again an adapted reduction of singularities of $\hat{\xi }$ with $\widetilde{\mathcal {D}}=(\mathcal {D}{\setminus } \{ \gamma _I\})\cup \{ \gamma _{I,-\infty }, \gamma _{I,\infty }, \gamma _{I,1} \}$ and $\widetilde{\mathcal {D}}^{nc}=(\mathcal {D}^{nc}{\setminus } \{\gamma _I\}) \cup \{ \gamma _{I,1} \}$. We will say that $\gamma _{I,1}$ emerges from $\gamma _I$. Moreover, the new non-corner characteristic cycle $\gamma _{I,1}$ is given by equations $\{ z^{(I,0)}=\rho ^{(I,0)}=0 \}$ in coordinates for which $\sigma _{\gamma _I}$ is written as

$$\begin{aligned} z^{(I)} = \rho ^{(I,0)}(z^{(I,0)}+a_{I,1}), \ \rho ^{(I)}= \rho ^{(I,0)}, \end{aligned}$$

where ${\hat{h}}_I'(0)=a_{I,1}$. We deduce that if $N\in \mathbb {N}$ and $j_N^{\rho ^{(I)}}({\hat{h}}_I)=j_N^{\rho ^{(I)}}(h_{\ell ,I}),$ then the strict transform $\sigma _{\gamma _{I}}^*{\hat{S}}_{\ell ,I}$ by $\sigma _{\gamma _{I}}$ is a formal surface along $\gamma _{I,1}$ and

$$\begin{aligned} \sigma _{\gamma _{I}}^{-1}({{\widetilde{\Sigma }}}_{N,\delta ,1}^{(\theta ,z^{(I)},\rho ^{(I)})} ( {\hat{S}}_{\ell ,I})) = {{\widetilde{\Sigma }}}_{N-1,\delta ,1}^{(\theta ,z^{(I,0)},\rho ^{(I,0)})} (\sigma _{\gamma _{I}}^*{\hat{S}}_{\ell ,I}). \end{aligned}$$

Lemma 6.2

Let $\varphi (\theta ,z,\rho )=(\theta + \varphi _\theta , z+ \varphi _z,\rho + \varphi _\rho )$ be a diffeomorphism along $\gamma =\{ z=0,\rho =0 \}$, where $\varphi _\theta ,\varphi _z,\varphi _\rho \in \mathbb {R}[\cos \theta , \sin \theta ]\{ z,\rho \}$ are of order at least two in $(z,\rho )$ and divisible by $\rho$. Let $S_i$, $i=1,2$, be formal surfaces with defining ideals generated by $F_i=z-f_i(\theta ,\rho )$ where $f_i(\theta ,\rho )\in \mathbb {R}[\cos \theta , \sin \theta ][[\rho ]]$ and such that $S_1=\varphi ^*(S_2)$, i.e. in terms of ideals $(F_1)=(F_2\circ \varphi )$. Then, for every cone ${{\widetilde{\Sigma }}}_{N,C_1,\delta _1}(S_1)$ with $\delta _1$ sufficiently small, there exist some constants $C_2,\delta _2>0$ such that:

$$\begin{aligned} \varphi ^{-1}(\widetilde{\Sigma }_{N,C_2,\delta _2}(S_2)) \subseteq \widetilde{\Sigma }_{N,C_1,\delta _1}(S_1). \end{aligned}$$

Fig. 5

${{\widetilde{\Sigma }}}_{N,C_1,\delta _1}(S_1)$ and $\varphi ^{-1}({{\widetilde{\Sigma }}}_{N,C_2,\delta _2}(S_2))$

See Fig. 5 for an illustration of this lemma.

Proof

Consider the functions $g_{i,\epsilon }(\theta ,z,\rho )=z-j_N^{\rho }(f_i(\theta ,\rho ))-\epsilon C_i\rho ^N$ for $i=1,2$ and $\epsilon \in \{ -1,+1 \}$. The boundary of the cone $\widetilde{\Sigma }_{N,C_i,\delta _i}(S_i)$ is given by three surfaces with equations $g_{i,+}=0$, $g_{i,-}=0$ and $\rho =\delta _i$. It is enough to prove that there exists $C_2,\delta _2>0$ such that the following holds.

• (i) The function $\rho \circ \varphi -\delta _2$ is positive in the points $K:=\{ \rho =\delta _1 \}\cap \overline{ {{\widetilde{\Sigma }}}_{N,C_1,\delta _1}(S_1) }$.

• (ii) The function $g_{2,+}\circ \varphi$ is positive in the set $\{ g_{1,+}=0, 0<\rho \circ \varphi <\delta _2 \}$.

• (iii) The function $g_{2,-}\circ \varphi$ is negative in the set $\{ g_{1,-}=0, 0<\rho \circ \varphi <\delta _2 \}$.

To get (i), we can take any $\delta _2< \inf ((\rho \circ \varphi )|_K)$, taking into account that K is compact and that $\rho \circ \varphi =\rho + \varphi _\rho$ only vanishes along the divisor $\rho =0$ in a neighborhood of $\mathbb {S}^1\times \{ 0\}$.

Notice that we have $z-j_{N} ^\rho (f_i)= F_i + O(\rho ^{N+1}), \ i=1,2,$ and $F_2\circ \varphi =F_1\cdot U$ where $U=1+T$ and $T\in \mathbb {R}[\cos \theta , \sin \theta ][[z,\rho ]]$ such that $\rho$ divides T. Then, we deduce

$$\begin{aligned} g_{2,+}\circ \varphi =(F_2-C_2\rho ^N + O(\rho ^{N+1}) )\circ \varphi = F_1\cdot U + ( -C_2\rho ^N+ O(\rho ^{N+1}))\circ \varphi . \end{aligned}$$

Using that $\rho \circ \varphi = \rho + \varphi _\rho =\rho (1+{\tilde{\varphi }}_\rho )$ with ${\tilde{\varphi }}_\rho \in \mathbb {R}[\cos \theta , \sin \theta ]\{ z,\rho \}$ of positive order, we have:

$$\begin{aligned} g_{2,+}\circ \varphi =F_1\cdot U -C_2\rho ^N +O(\rho ^{N+1}). \end{aligned}$$

Now, we evaluate $g_{2,+}\circ \varphi$ on the sets $\{g_{1,+}=0\}$, in other words, when $z=j_{N}^\rho (f_1)+ C_1\rho ^N$. Considering that $F_1(\theta ,j_{N}^\rho (f_1)+ C_1\rho ^N,\rho )=f_1+O(\rho ^{N+1})+C_1\rho ^N -f_1= C_1\rho ^N+O(\rho ^{N+1})$, we get:

$$\begin{aligned} & g_{2,+}\circ \varphi (\theta ,j_{N}^\rho (f_1)+ C_1\rho ^N,\rho )\\ & \quad = (C_1\rho ^N+O(\rho ^{N+1})) \cdot (1+T(\theta ,j_{N}^\rho (f_1)+ C_1\rho ^N,\rho )) -C_2\rho ^N + O(\rho ^{N+1}). \end{aligned}$$

Since $\rho$ divides T, we obtain finally

$$\begin{aligned} g_{2,+}\circ \varphi (\theta ,j_{N}^\rho (f_1)+ C_1\rho ^N,\rho )=(C_1-C_2)\rho ^N +O(\rho ^{N+1}). \end{aligned}$$

Thus, taking $C_2<C_1$, we have that $g_{2,+}\circ \varphi (\theta ,j_{N}^\rho (f_1)+ C_1\rho ^N,\rho )>0$ for $\rho$ small enough. We get (ii), up to taking a smaller $\delta _2$. The arguments to get (iii) are similar. $\square$

Denote $\mathcal {D}^{nc,fix}=\{ \gamma _1,\ldots , \gamma _r \}$ and $\mathcal {D}^{nc,nfix}=\{ \gamma _{r+1},\ldots , \gamma _s \}$ and, for any j, let $I_j\in \mathcal {I}$ be defined by $\gamma _j=\gamma _{I_j}$. Let $J_j\in \mathcal J$ be the index of the chart $(C_{J_j}, (\theta ,z^{(J_j)},\rho ^{(J_j)}))$ as presented in the beginning of Sect. 5.3, where the cycle $\gamma _{I_j}$ is given by $z^{(J_j)}=\rho ^{(J_j)}=0$. Denote also $N_j=N_{\ell ,I_j}$.

Consider the sequence of admissible blowing-ups $\mathcal { M}'=(M',\pi ',\mathcal {A}',\mathcal {D}')$ over $\mathcal {M}$ constructed as follows. For each $j=1,\dots , s$, let $\tau _j$ be the composition

$$\begin{aligned} \tau _j=\sigma _{j,1}\circ \cdots \circ \sigma _{j,N_j}, \end{aligned}$$

where $\sigma _{j,1}$ is the admissible blowing-up whose center is the characteristic cycle $\gamma _j=\gamma _{I_j}$, $\sigma _{j,2}$ is the admissible blowing-up whose center is the non-corner characteristic cycle $\gamma _{I_j,1}$ emerging form $\gamma _{I_j}$ (cf. Remark 6.1), and so on. Then, $\mathcal M'$ is the resulting sequence of admissible blowing-ups by setting $\pi '=\pi \circ \tau _1 \circ \cdots \circ \tau _s$.

Notice that $\mathcal M'$ is an adapted reduction of singularities of $\hat{\xi }$ with the same number of non-corner characteristic cycles as $\mathcal M$. We put ${\mathcal {D}'}^{nc}=\{ \gamma _1',\cdots , \gamma _s' \}$ where $\gamma _j'$ emerges from $\gamma _j$ by the composition of $\tau _j$. Now, we take $\ell '\in \mathbb {N}$ satisfying $\ell '\ge \max \{ \ell , \ell _{\mathcal { M}'}+1 \}$.

Proposition 6.3

The vector field $\xi _{\ell '}$ satisfies Theorem 1.1.

Proof

Choose $0<\delta '\le \delta$ sufficiently small and an open set $V_j$ with ${{\widetilde{\Sigma }}}_{N_j,\delta ',1}({\hat{S}}_{\ell ,I_j})\subset V_j \subset {{\widetilde{\Sigma }}}_{N_j,\delta ,1}({\hat{S}}_{\ell ,I_j})$ for any $j=1,\ldots , s$ so that they satisfy

• (a) The Poincaré map $P_{\ell ,I_j}$ is defined in ${{\widetilde{\Sigma }}}_{N_j,\delta ,1}({\hat{S}}_{\ell ,I_j})\cap \{\theta =0\}={\Sigma }_{N_j,\delta }(\Gamma _{\ell ,I_j})$ and satisfies there the conclusions of Lemmas 5.7 or 5.8, correspondingly.

• (b) If Z is a cycle of the transform $\xi _{\ell }^{(J_j)}$ contained in $V_j$, it intersects $\{ \theta =0 \}$.

• (c) If $j\in \{ 1,\ldots ,r \}$ then $\Gamma _{\ell ,I_j}\subset \text {Fix}(P_{\ell ,I_j})$ admits a representative in ${{\widetilde{\Sigma }}}_{N_j,\delta ,1}({\hat{S}}_{\ell ,I_j})$ denoted again $\Gamma _{\ell ,I_j}$ whose intersection with $V_j\cap \{\theta =0\}$ is a connected analytic regular curve.

• (d) For any $a\in \Gamma _{\ell ,I_j}\cap V_j$ when $j\in \{ 1,\ldots ,r \}$, the cycle of $\xi _{\ell }^{(J_j)}$ through a is contained in $V_j$.

The existence of these objects with such properties is guaranteed by standard arguments using the continuity of the flow of $\xi _{\ell }^{(J_j)}$ and the fact that each $\gamma _j$ is a cycle of $\xi _{\ell }^{(J_j)}$. Notice that if $j\in \{ r+1,\ldots , s\}$ we can take $\delta '=\delta$ and $V_j$ to be equal to the solid cone ${{\widetilde{\Sigma }}}_{N_j,\delta ,1}({\hat{S}}_{\ell ,I_j})$

Define, for $j=1,\ldots ,r$, the set $\widetilde{S}_j$ given by the saturation of $\Gamma _{\ell ,I_j}{\cap V_j}$ by the flow of $\xi _{\ell }^{(J_j)}$. By the above properties, ${{\widetilde{S}}}_j$ is an analytic submanifold of $V_j\subset M$, intersecting the divisor $\pi ^{-1}(0)$ along $\gamma _j$ and completely filled up with cycles of $\xi _{\ell }^{(J_j)}$. We have, furthermore from (a):

$$\begin{aligned} \mathcal {C}_{ \bigcup _{j=1}^s V_j} (\pi ^*\xi _{\ell }) ={{\widetilde{S}}}_1\cup \cdots \cup {{\widetilde{S}}}_r. \end{aligned}$$

(51)

With the notations of Eq. (3), the diffeomorphism $\psi _{\ell ,\ell '}:=\psi _{\ell '}\circ \psi _{\ell }^{-1}$ is analytic and conjugates $\xi _\ell$ and $\xi _{\ell '}$, namely $\xi _{\ell }=\psi _{\ell ,\ell '}^*\xi _{\ell '}$. Moreover, since $\ell '\ge \ell \ge \ell _{\mathcal { M}}+1$, we have that $j_\ell (\psi _{\ell ,\ell '})=Id$ and we can apply Proposition 3.16 to $\psi _{\ell ,\ell '}$. We obtain an analytic conjugation $\psi _{\ell ,\ell '}^{(J_j)}$ between $\xi _{\ell }^{(J_j)}$ and $\xi _{\ell '}^{(J_j)}$ in a neighborhood of $\gamma _j$. Up to shrinking $\delta$ and $\delta '$, we may assume that $\psi _{\ell ,\ell '}^{(J_j)}$ is well defined and one-to-one in $V_j$ for any j. Moreover, $\psi _{\ell ,\ell '}^{(J_j)}$ is in the conditions of Lemma 6.2 with respect to the coordinates $(\theta , z^{(J_j)},\rho ^{(J_j)})$ and satisfies ${\hat{S}}_{\ell ,I_j}=(\psi _{\ell ,\ell '}^{(J_j)})^*( {\hat{S}}_{\ell ',I_j})$. Let $W_j:=\psi _{\ell ,\ell '}^{(J_j)}(V_j)$ for $j=1,\ldots ,r$. Using the conclusion of Lemma 6.2 and the fact that ${{\widetilde{\Sigma }}}_{N_j,\delta ',1}({\hat{S}}_{\ell ,I_j})\subset V_j$ for any j, we have that each $W_j$ has a solid cone around ${\hat{S}}_{\ell ',I_j}$ of the form ${{\widetilde{\Sigma }}}_{N_j,\delta ',1}({\hat{S}}_{\ell ',I_j})$ (same order $N_j$).

We deduce from (51) and by conjugation the following:

$$\begin{aligned} \mathcal {C}_{ \bigcup _{j=1}^s W_j} (\pi ^*\xi _{\ell '}) = {{\widetilde{S}}}_1 '\cup \cdots \cup {{\widetilde{S}}}_r', \end{aligned}$$

(52)

where ${{\widetilde{S}}}_j':=(\psi _{\ell ,\ell '}^{(J_j)})({\widetilde{S}}_j)\subset W_j$, for $j=1,\ldots , r$. By Remark 6.1 and taking into account that $\ell \ge l + N_j +1$ for any j, we have that $W_j':=\tau _j^{-1}(W_j)$, together with the intersection of its closure with the divisor $\tau _j^{-1}(\gamma _j)$, is a neighborhood of $\gamma _j'$ in ${M}'$. We may assume that $W_j'\cap W'_k=\emptyset$ if $j\ne k$. Complete the union $\displaystyle \bigcup _{j=1}^s W_j'$ to a neighborhood $W'$ of $\text {Supp}(\mathcal {D}')$ in $M'$ adding two by two disjoint neighborhoods of the elements $\gamma \in \mathcal {D'}{\setminus } {\mathcal {D}'}^{nc}$ where, correspondingly, Propositions 5.1 or 5.2 holds. We apply finally Proposition 4.1 to $W'$ (recall $\ell '\ge \ell _{\mathcal { M}'}+1$): there is a neighborhood U of $0\in \mathbb {R}^3$ such that $(\pi ')^{-1}(\mathcal {C}_U(\xi _{\ell '})) \subset W'$. By Propositions 5.1 or 5.2 we get moreover that

$$\begin{aligned} (\pi ')^{-1}(\mathcal {C}_U(\xi _{\ell '})) \subset \displaystyle \bigcup _{j=1}^s W_j'. \end{aligned}$$

This equation, together with (52) shows that, if we put $S_j:=\pi ({{\widetilde{S}}}_j')\cap U$ for $j=1,\ldots ,r$, then

$$\begin{aligned} \mathcal {C}_U(\xi _{\ell '})\subset S_1\cup \cdots \cup S_r. \end{aligned}$$

(53)

Notice that each $S_j$ is an analytic smooth surface in $U{\setminus } \{ 0\}$ and that $0\in {\bar{S}}_j$. Up to taking a smaller U, we may assume that ${\bar{S}}_j\cap U=S_j\cup \{ 0 \}$. Moreover, $S_j$ is a subanalytic set, since $\pi$ is proper and ${{\widetilde{S}}}_j'$ is semi-analytic. Finally, by construction, we have that $S_j\subset \pi ( { {{\widetilde{S}}}_j' } )$ for $j=1,\ldots ,r$, the two sets have the same germ at $0\in \mathbb {R}^3$ and the later is entirely composed of cycles for every $j=1,\ldots ,r$. This implies, together with (53) that, if $V\subset U$ is any open neighborhood of $0\in \mathbb {R}^3$ such that $\text {Fr}(V)\cap S_j$ coincides with one of such cycles for every $j=1,\ldots ,r$, then we have

$$\begin{aligned} \mathcal {C}_V(\xi _{\ell '})=(S_1\cup \cdots \cup S_r) \cap V. \end{aligned}$$

This ends the proof. $\square$