Chaotic Dynamics at the Boundary of a Basin of Attraction via Non-transversal Intersections for a Non-global Smooth Diffeomorphism

Abstract

In this paper, we give analytic proofs of the existence of transversal homoclinic points for a family of non-globally smooth diffeomorphisms having the origin as a fixed point which come out as a truncated map governing the local dynamics near a critical period three-cycle associated with the Secant map. Using Moser’s version of Birkhoff–Smale’s theorem, we prove that the boundary of the basin of attraction of the origin contains a Cantor-like invariant subset such that the restricted dynamics to it is conjugate to the full shift of N-symbols for any integer $N\ge 2$ or infinity.

1 Introduction

The question of whether a dynamical system admits invariant subsets of the phase portrait in which the dynamics is chaotic goes back to the origins of this area of mathematics. Studying the existence, or not, of chaotic dynamics and determine the topology and the geometry of the subsets where this happens has become a classical problem. Nevertheless, since the question arises in so many distinct scenarios, there has been different approaches to this phenomena, including the use of non-equivalent mathematical definitions in order to capture the meaning of chaos in each particular case. Measuring chaos in high, or even infinite, dimensional Hamiltonian dynamical systems or doing so for one-dimensional interval dynamics requires to particularize the meaning of the word chaos to concrete mathematical definitions.

Nonetheless, once we agree on which dynamical properties characterize chaos (density of periodic points, transitivity, dense orbits, sensibility with respect to initial conditions, all at once,...), a common accepted approach to ensure chaotic dynamics is to show that, in certain dynamically invariant region(s) of our phase portrait, the dynamics is conjugate (that is, equal up to a homeomorphism) to the one of a model for which it is somehow easy to test the properties mentioned above.

The usual toy model is the dynamical system $(\Sigma _N,\sigma )$, where $\Sigma _N$ is the set of bi-infinite (or one-side) sequences of $N\ge 2$ symbols and $\sigma$ is the shift map; see (Moser 2001). One can easily check that the system $(\Sigma _N,\sigma )$ captures the dynamical properties proposed above. Since the conjugacy sends orbits of our dynamical systems to orbits of the shift map acting on the space of symbols, this methodology is also known as symbolic dynamics. To focus on the content of this paper and simplify the discussion, let us assume we have a discrete dynamical system in ${\mathbb {R}}^2$ generated by the iterates of a (smooth) map.

In any event, the difficult part to apply this strategy is to show that in some regions of the phase portrait our dynamics is conjugate to the dynamical system $(\Sigma _N,\sigma )$. A major result in this direction goes back to the cornerstone ideas of S. Smale (Birkhoff–Smale’s theorem) and J. Moser Moser (2001) who provide checkable (in some cases only numerically) dynamical conditions to ensure that a given dynamical system has a subset of the phase portrait whose dynamics is conjugated to the full-shift of an arbitrary number of symbols (even infinitely many). Roughly speaking they showed that if a smooth map has a transversal homoclinic intersection between the stable and unstable invariant manifolds of a hyperbolic saddle fixed point then, there is an invariant Cantor set whose restricted dynamics is conjugate to $(\Sigma _N,\sigma )$.

Even though the results have been extremely helpful in many different contexts (and extended in many different directions), we emphasize that the hypotheses include three key ingredients: the hyperbolicity of the saddle point, the map is a global diffeomorphism and the transversality of the intersection of the invariant manifolds. The main goal of this paper is to address the presence of chaotic dynamics, for a concrete family of maps, under the lack of two of the conditions; the inverse map would not be globally smooth and in a first step we only can prove (analytically) that we have an intersection with a finite order contact.

Concretely, in this paper we consider the map

$$\begin{aligned} T_{d}\left( \begin{array}{l} x \\ y \end{array} \right) = \left( \begin{array}{l} y - (x+y)^d \\ y - 2(x+y)^d \end{array} \right) , \end{aligned}$$

(1.1)

with $d\ge 3$ being odd. Such map is a truncated expression of the third iterate of the (extended) Secant map applied to a polynomial p(x) near a critical period three-cycle

$$\begin{aligned} (c,c) \mapsto (c,\infty ) \mapsto (\infty ,0) \mapsto (c,c), \end{aligned}$$

where $p^{\prime }(c)=0$ (but $p(c)\ne 0$). See (Bedford and Frigge 2018; Garijo and Jarque 2019, 2022; Fontich et al. 2024) for more details. For later discussions we point out here that $T_d$ is a global homeomorphism, but it is not a global diffeomorphism since the inverse map, $T_d^{-1}$, is not smooth over the line $\{y=x\}$ (see (2.1) for its particular expression).

One can easily check that the origin of (1.1) is a fixed point and its basin of attraction

$$\begin{aligned} {\mathcal {A}}_{d}(0) = \{ (x,y) \in {\mathbb {R}}^2\, | \ T_{d}^n(x,y) \rightarrow (0,0) \, \text { as } \, n \rightarrow \infty \} \end{aligned}$$

(1.2)

is not empty. In Fontich et al. (2024) we proved the following topological description of ${\mathcal {A}}_{d}(0)$ and further information about its boundary. We denote $p_0=(0,1)$ and $p_1=(0,-1)$.

Theorem 1.1

Let $d\ge 3$ odd. Then, ${\mathcal {A}}_{d}(0)$ is an open, simply connected, unbounded set. Moreover, $\partial {\mathcal {A}}_{d}(0)$ contains the stable manifold of the hyperbolic two-cycle $\{p_0,p_1\}$ lying in $\partial {\mathcal {A}}_{d}(0)$.

Fig. 1

The picture (in red) of the set ${\mathcal {A}}_{3}(0)$. Notice that according to Theorem 1.1 the red region is connected, simply connected and unbounded (Color figure online)

The thesis of the above theorem glimpse the possible topological complexity of $\partial {\mathcal {A}}_{d}(0)$ (see Fig. 1). In fact, the main goal of this paper is to provide a better understanding of $\partial {\mathcal {A}}_{d}(0)$ by proving that, apart from the stable manifold of the hyperbolic two-cycle $\{p_0,p_1\}$ there is a Cantor subset of $\partial {\mathcal {A}}_{d}(0)$ where the dynamics is conjugated to the one of the shift of N symbols and so inhering all its chaotic dynamics.

In Fontich et al. (2024), we were able to describe and bound the shape of a piece of the unstable manifold of $p_0$ for $T^2_d$ (and it was a key point in the arguments to prove Theorem 1.1). In this paper, we mimic some of the arguments there to control the shape of a piece of the stable manifold of $p_1$. Using both constructions and a singular $\lambda$-Lemma (Rayskin 2003), we can ensure the existence of homoclinic (not necessarily linearly transversal) points for $T_d^2$.

Theorem A

Let $\{p_0,p_1\}$ be the hyperbolic two cycle lying in the boundary of $\partial {\mathcal {A}}_{d}(0)$. Then, the stable and unstable manifolds of $p_1$ (as well as $p_0$), as a fixed point for $T_d^2$, intersect at a homoclinic point.

Going back to our previous arguments, if we want to apply Birkhoff–Smale’s theorem, we need to prove the existence of transversal homoclinic points, so that Theorem A is not enough. In Churchill and Rod (1980), the authors are able to conclude transversal intersections under the presence of (topological) homoclinic intersections, but their map is area preserving, is a global smooth diffeomorphism and admits, in a sufficiently small neighbourhood the hyperbolic saddle, a concrete local normal form which provides a first integral.

In our case, we have not the previously mentioned normal form and since $T_d^{-1}$ is not smooth over the line $\{y=x\}$ we cannot use that the globalization of the stable manifold of the 2-cycle $\{p_0,p_1\}$ by applying $T_d^{-2}$ is analytic. In any event, inspired in the strategy proof in Churchill and Rod (1980), using alternative arguments to deal with our weaker conditions we are able to conclude the existence of transversal homoclinic intersections.

Theorem B

Let $\{p_0,p_1\}$ be the hyperbolic two cycle lying in the boundary of $\partial {\mathcal {A}}_{d}(0)$. Then, the stable and unstable manifolds of $p_1$ (as well as $p_0)$, as a fixed point for $T_d^2$, intersect transversally.

Although from the previous theorem, we have the existence of transversal homoclinic points we still cannot directly apply Birkhoff–Smale’s theorem since the inverse map, $T_d^{-1}$ map is not globally smooth. However, we can overcome this difficulty and prove the main result of this paper.

Theorem C

There exists an invariant Cantor set, contained in $\partial \mathcal {A}_{a,d}(0)$, where the dynamics of $T_d^2$ is conjugate to the full shift of N-symbols. In particular, $\partial \mathcal {A}_{a,d}(0)$ contains infinitely many periodic points with arbitrary high period.

Fig. 2

This picture corresponds to $d=3$. In red we plot the attracting basin ${\mathcal {A}}_3(0)$. In blue (respectively, yellow) we draw the stable (respectively, unstable) manifold of the two cycle $\{p_0,p_1\}$. The picture illustrates (numerically) the transversal intersections described in Theorem B. According to Theorem C, $\partial \mathcal {A}_{d}(0)$ contains the stable manifold of the two cycle (in blue) and a Cantor-set like with chaotic dynamics (Color figure online)

We emphasize that the theorems provide analytic proofs, rather than numerical evidence, of non-local properties of invariant manifolds for a family of maps. There are few cases where this has been done. For instance, in Fontich (1990), there is an analytical proof of the transversal intersection of the invariant manifolds for a wide range of a parameter for a class of maps which include the conservative Hénon map and the Chirikov standard map. In Gelfreich (1999), there is an analytical proof of the transversal intersection of the invariant manifolds of the standard map when the angle is exponentially small with respect to the parameter of the family. Also, in Delshams and Ramírez-Ros (1996) and Martín et al. (2011) they prove transversal intersection for the manifolds of close to integrable maps.

We organize the paper as follows. In Sect. 2, we summarize some preliminaries from Fontich et al. (2024) that we need in the proofs of the present paper, trying to make the present paper self-contained. In Sect. 3 we prove Theorem A, in Sect. 4 we prove Theorem B and finally in Sect. 5 we conclude the proof Theorem C.

2 Preliminaries

In this section, we collect some preliminary results about the map $T_d:\mathbb {R}^2\rightarrow \mathbb {R}^2$, introduced in (1.1), for $d\ge 3$, an odd number. Everything was already introduced in Fontich et al. (2024), but for the sake of completeness and easier reading, we include them here.

The map $T_d:\mathbb {R}^2\rightarrow \mathbb {R}^2$ is a polynomial and a homeomorphism and its inverse map is real analytic in $\mathbb {R}^2 \setminus \{x=y\}$, but not differentiable on the line $\{x=y\}$. Its inverse is given by

$$\begin{aligned} T^{-1}_{d} (x,y) = \left( -2x+y + \left( x-y\right) ^{1/d}, \, 2x-y \right) . \end{aligned}$$

(2.1)

Observe that $T_d^{-1}(x,x)=(-x,x)$ for all $x \in {\mathbb {R}}$. One can easily check that $T_d$ has a unique two-cycle $\{p_0=(0,1), p_1=(0,-1)\}$, i.e. $p_1= T_d(p_0)$ and $p_0= T_d(p_1)$. This two-cycle will play a fundamental role in the dynamics of $T_d$. Moreover, we have that

$$\begin{aligned} DT_d^2(p_0)=DT_d^2(p_1)= DT_d(p_0) DT_d(p_1) = \left( \begin{array}{ll} 3d^2-2d & \ \ 3d^2-4d+1 \\ 6d^2-2d & \ \ 6d^2-6d+1 \end{array} \right) .\qquad \end{aligned}$$

(2.2)

A direct computation shows that the characteristic equation of $DT_d^2(p_0)$ is

$$\begin{aligned} p(\lambda )= \lambda ^2 -(1- 8d + 9 d^2)\lambda + d^2 =0 \end{aligned}$$

and the eigenvalues and eigenvectors are given by

$$\begin{aligned} \lambda ^{\pm }_d = \frac{1}{2}\left( 9d^2- 8d + 1 \pm (3d-1)\sqrt{ 9 d^2 -10d + 1} \right) \end{aligned}$$

(2.3)

and

$$\begin{aligned} (1,m^{\pm }_d) = \left( 1,\ \frac{ 4d}{1-d \pm \sqrt{ 9d^2-10d + 1}} \right) , \end{aligned}$$

(2.4)

respectively.

On the one hand, it is easy to check that both eigenvalues are strictly positive. Moreover, $\lambda ^{-}_d$ is strictly decreasing and $\lambda ^+_d$ is strictly increasing, with respect to the parameter d. We also have

$$\begin{aligned} \begin{array}{lll} \lim \limits _{d\rightarrow \infty } \lambda ^{-}_d=1/9 & \quad \text {and} & \quad 1/9<\lambda ^{-}_d \le \lambda ^{-}_3 = 29-8\sqrt{13} \approx 0.1556 \\ \lim \limits _{d\rightarrow \infty } \lambda ^{+}_d=\infty & \quad \text {and} & \quad \lambda ^{+}_ d \ge \lambda ^{+}_ 3 = 29+8\sqrt{13}\approx 57.8444. \end{array} \end{aligned}$$

On the other hand, $m^{-}_d$ is negative and strictly increasing while $m^+_d$ is positive and strictly decreasing (both with respect to the parameter d). We also have

$$\begin{aligned} \begin{array}{ccl} \lim \limits _{d\rightarrow \infty } m^{-}_d =-1 & \quad \text {and} & \quad -1.3028 \approx \frac{-6}{1+\sqrt{13}} = m^{-}_3 \le m^{-}_d<-1, \\ \lim \limits _{d\rightarrow \infty } m^{+}_d =2 & \quad \text {and} & \quad 2<m^{+}_d \le m^{+}_3 = \frac{6}{\sqrt{13}-1} \approx 2.3028. \end{array} \end{aligned}$$

Therefore, the two cycle $\{p_0, p_1\}$ is hyperbolic of saddle type. We denote $W^s_{p_j}$, $W^u_{p_j}$ the stable and the unstable manifolds of the fixed points $p_j$ for the map $T_d^2$, $j=0,1$. Similarly we denote by $W^s_{\textrm{loc},\, p_j}$, $W^u_{\textrm{loc},\, p_j}$ the corresponding local stable and unstable manifolds of some size $\delta$ that we do not make explicit in the notation. Actually, given some size $\delta >0$,

$$\begin{aligned} W^s_{\textrm{loc},\, p_j} = \{ z\in \mathbb {R}^2\mid \, T^{2k}_d (z) \in B_\delta (p_j) \ \ \text { for all} \ \ k\ge 0 \}, \end{aligned}$$

where $B_{\delta }(p_j)$ denotes the open ball centred at $p_j$ with radius $\delta >0$, for $j=0,1$. We define analogously $W^u_{\textrm{loc},\, p_j}$ for $T^{-2k}_d$.

We also denote

$$\begin{aligned} W^s:=W^s_{\{p_0,p_1\}} \quad \textrm{and} \quad W^u:=W^u_{\{p_0,p_1\}}\end{aligned}$$

the global stable and unstable manifolds of the periodic orbit $\{p_0,p_1\}$, respectively. Since $T_d$ is analytic on ${\mathbb {R}}^2$ and $T^{-1}_d$ is analytic on ${\mathbb {R}}^2 {\setminus } \{y=x\}$ the local versions of the invariant manifolds are analytic. Moreover, the (global) unstable manifold, obtained iterating by $T_d$ the local one, is analytic and the (global) stable manifold, obtained iterating by $T^{-1}_d$, is analytic except at the preimages of the intersections of $W^s$ with $\{y=x\}$.

When there is no confusion we use the simplified notation $\lambda ^{\pm }:= \lambda ^{\pm }_d$ and $m^{\pm }:= m^{\pm }_d$.

2.1 The triangle ${\mathcal {D}}$ and its images: $T_d({\mathcal {D}})$ and $T_d^{-1}({\mathcal {D}})$.

In Fontich et al. (2024, Section 5) we considered the triangle ${\mathcal {D}}$ of vertices

$$\begin{aligned} p_1=(0,-1), \qquad \left( \frac{1}{ m^+ +1},\frac{-1}{ m^+ +1}\right) \qquad \text{ and } \qquad \left( \frac{1}{m^\star +1},\frac{-1}{m^\star +1}\right) , \end{aligned}$$

where $m^\star = 7/2$, or equivalently,

$$\begin{aligned} {{\mathcal {D}}} = \{ (t,-1+mt) \, | t \in [0,1/(m+1)], m \in [m^+, m^{\star }] \}. \end{aligned}$$

We also considered the sets $T_d({\mathcal {D}})$ and $T^{-1}_d({\mathcal {D}})$. We showed that the set $T_d({\mathcal {D}})$ is bounded by the images of the sides of ${\mathcal {D}}$ given by the curves $\gamma _{m^+}(t)$, $\gamma _{ m^\star }(t)$ where

$$\begin{aligned} \gamma _{m}(t) \!=\! T_d (t,-1+m t) = ( mt -1 - ( (m+1)t -1)^d, mt -1 \!-\! 2( (m\!+\!1)t -1)^d),\nonumber \\ \end{aligned}$$

(2.5)

for $0\le t\le \frac{1}{m+1}$, and

$$\begin{aligned} \partial T_d({\mathcal {D}}) \cap \{y=x\}=\left\{ (t,t) \mid \ \frac{-1}{ m^++1} \le t \le \frac{-1}{m^\star +1}\right\} . \end{aligned}$$

Finally, we claim that there is a (connected) piece of $W^u_{p_0} \cap \{y\le 1\}$, tangent to the line $y=1+m^{+}x$ at $p_0$, contained in $T_d({\mathcal {D}})$ joining the point $p_0$ with some point in $\partial T_d({\mathcal {D}}) \cap \{y=x\}$. We call left and right boundaries of $T_d({\mathcal {D}})$ the curves $\gamma _{m^+}(t)$ and $\gamma _{m^\star }(t)$, respectively. See Fig. 3 (left). We do not include here the arguments used in Fontich et al. (2024, Lemma 5.4) to prove the claim but in the next section we mimic, including all computations, the ideas used in Fontich et al. (2024) for the case of $\widehat{{\mathcal {D}}}$, $T_d(\widehat{{\mathcal {D}}})$ and $T_d^{-1}(\widehat{{\mathcal {D}}})$.

3 Proof of Theorem A

To prove Theorem A, we first show the existence of an heteroclinic intersection for the map $T_d$. More precisely, we have the following statement.

Proposition 3.1

Let $\{p_0,p_1\}$ be the hyperbolic two-cycle lying in the boundary of $\partial {\mathcal {A}}_{d}(0)$ (see Theorem 1.1). Then, the unstable manifold of $p_0$ and the stable manifold of $p_1$ intersect in a heteroclinic point.

The idea is to show that $T_d({\mathcal {D}})$ (Fig. 3, left) and $T_d^{-1}(\widehat{{\mathcal {D}}})$ ((Fig. 3, right) intersect in a suitable manner that forces the intersection of the invariant manifolds (Fig. 4). Since the proof of this proposition is quite long, we split it into several lemmas. We assume all notation introduced in Sect. 2. In particular, we have described the construction provided in Fontich et al. (2024, Lemma 5.4) to localize the piece of the unstable manifold attached to $p_0$ inside $T_d({{\mathcal {D}}})$. The first step is to make a similar construction to localize a piece of the stable manifold of $p_1$. Let

$$\begin{aligned} -\frac{3}{2}\le \widehat{m}^{\star }= \widehat{m}_d^{\star }:= -1-\frac{1}{d-1} = \frac{-d}{d-1}< m^{-}<-1, \end{aligned}$$

(3.1)

where the inequalities follow from direct computations. We introduce the triangle $\widehat{{\mathcal {D}}}$ with vertices

$$\begin{aligned} p_0=(0,1), \qquad \left( \frac{-1}{m^- +1},\frac{-1}{m^- +1}\right) \qquad \text{ and } \qquad \left( \frac{-1}{\widehat{m}^\star +1},\frac{-1}{\widehat{m}^\star +1}\right) , \end{aligned}$$

or equivalently,

$$\begin{aligned} \widehat{\mathcal {D}}= \{ (t,1+mt) \, | \ t \in [0,1/(1-m)], \ m \in [\widehat{m}^\star , {m^-}]\}. \end{aligned}$$

Fig. 3

Left: The triangle ${\mathcal {D}}$, its image $T_d({\mathcal {D}})$ and (dashed, red) a piece of $W_{p_0}^u$ attached to $p_0$. Right: The triangle $\widehat{\mathcal {D}}$, its images $T_d({\mathcal {D}})$ and $T_d^{-1}(\widehat{{\mathcal {D}}})$, and (dashed, blue) a piece of $W_{p_1}^s$ attached to $p_1$. We also add the relevant objects appearing in the proof of Proposition 3.1 and Theorem A (Color figure online)

As we did with the set ${\mathcal {D}}$ in Fontich et al. (2024, Lemma 5.3), we study the geometry of the sets $T_d(\widehat{\mathcal {D}})$ and $T_d^{-1}(\widehat{{{\mathcal {D}}}})$. From the properties of these sets, we will prove that there is a piece of $W^s_{p_1}\cap \{ y\ge -1\}$ that is contained in $T_d^{-1}(\widehat{{\mathcal {D}}})$. Moreover, this piece joints $p_1$ with a point in $T_d^{-1}(\widehat{{\mathcal {D}}}) \cap \{y=-x\}$. See the right picture in Fig. 3. Then, using the geometry of the intersection of $T_d({\mathcal {D}})$ and $T_d^{-1}(\widehat{{\mathcal {D}}})$ we will prove that $W^u_{p_0}$ and $W^s_{p_1}$ have to cross (topologically) in a heteroclinic intersection, proving Proposition 3.1. From this heteroclinic intersection, we will obtain a homoclinic intersection as claimed in Theorem A.

The preimage $T_d^{-1}(\widehat{{\mathcal {D}}})$. We denote by $\widehat{\Gamma }_m(t)$ the image by $T_d^{-1}$ of the segment $\{(t,1+mt)\mid t\in [0,1/(1-m)] \}$. Thus,

$$\begin{aligned} \widehat{\Gamma }_m(t) = T_d^{-1}(t,1+mt)=: (\widehat{\alpha }_m(t),\widehat{\beta }_m(t)), \end{aligned}$$

(3.2)

where

$$\begin{aligned} \widehat{\alpha }_m(t) = (m-2)t +1 + ( (1-m)t -1)^{1/d} \quad \textrm{and} \quad \widehat{\beta }_m(t) = (2-m)t -1. \end{aligned}$$

We are interested in $\widehat{\Gamma }_m(t)$ for $m \in [\widehat{m}^\star ,m^-]$. Note that the point on $\widehat{\mathcal {D}}\cap \{y=x\}$ corresponds to $t= 1/(1-m)$ and is mapped by $T_d^{-1}$ to $(-1/(1-m), 1/(1-m))$ on the line $\{y=-x\}$. Taking derivatives, we have that

$$\begin{aligned} \widehat{\alpha }_m'(t) = m-2 + \frac{1-m}{d} ((1-m)t -1)^{(1-d)/d}, \ \ \widehat{\beta }_m'(t) = 2-m>0, \ \ \widehat{\alpha }_m''(t)>0 \ \ \textrm{and} \ \ \widehat{\beta }_m''(t)=0. \end{aligned}$$

A direct computation shows that $\widehat{\alpha }'_m(t)=0$ if and only if $t= t_{\pm }$, where

$$\begin{aligned} t_{\pm } = \frac{1}{1-m}\left( 1 \pm \left( \frac{1-m}{d(2-m)}\right) ^{d/(d-1)}\right) \end{aligned}$$

and $0<t_-< \frac{1}{1-m} < t_+$, since, as $m<0$, we have $0<\frac{1-m}{d(2-m)} < 1$. It follows from these computations that $\widehat{\alpha }_m(t),\ m \in [\widehat{m}^\star ,m^-],$ has a unique minimum (in its domain) at $t_- \in (0, \frac{1}{1-m})$. Finally, $\widehat{\alpha }'_m( \frac{1}{1-m})=\infty$ which means that when $\widehat{\Gamma }_m(t)$ meets $\{y=-x\}$, its tangent line is horizontal. See the right picture in Fig. 3. In other words the vectors $\widehat{\Gamma }_m'( \frac{1}{1-m})$ and $\widehat{\Gamma }_m'(t_-)$ are parallel to the lines $y=0$ and $x=0$, respectively.

Since $\widehat{\beta }_m(t)$ is invertible (linear), for any m we can represent the curve $\widehat{\Gamma }_m(t)$ as the graph of a function $x=g(y),\ y\in [-1,1/(1-m)]$ (remember that $1/(1-m)>0$), by taking $g(y) = \widehat{\alpha }_m \circ \widehat{\beta }^{-1}_m(y)$. Since $\widehat{\beta }_m''(t)=0$, we have that

$$\begin{aligned} & \frac{\text {d} g}{\text {d} y}(y)= \left[ \frac{\text {d} \widehat{\alpha }_m}{\text {d} t} \left( \frac{\text {d} \widehat{\beta }_m}{dt}\right) ^{-1}\right] \circ \widehat{\beta }^{-1}_m(y) \qquad \textrm{and} \\ & \quad \frac{\text {d} ^2\,g}{\text {d} y^2}(y)= \frac{\text {d} ^2 \widehat{\alpha }_m}{\text {d} t^2} \left( \frac{\text {d} \widehat{\beta }_m}{\text {d} t}\right) ^{-2} \circ \widehat{\beta }^{-1}_m(y)>0. \end{aligned}$$

The convexity of g implies that the image of $\widehat{\Gamma }_m(t)$ is above its tangent line at $p_1= (0,-1)$. In case $m= m^-$, this tangent line has slope $m^-$ and it is the minimum slope for all $m\in [\widehat{m}^{\star },{m}^-]$. Therefore, $T_d^{-1}(\widehat{{\mathcal {D}}})$ is above the line $y=m^- x-1$.

Also, g has a unique minimum at $y_-=\widehat{\beta }_m(t_-)$. Moreover, $\widehat{\Gamma }_m(t)$ intersects $\{y=0\}$ when $t= 1/(2-m)$ at the point $(x,y)=(\widehat{\alpha }_m (1/(2-m)),0)$ with

$$\begin{aligned} \widehat{\alpha }_m (1/(2-m)) = -\left( \frac{1}{2-m}\right) ^{1/d}. \end{aligned}$$

Again, the convexity of the function g implies that its graph intersected with $\{y\le 0\}$ is below the line

$$\begin{aligned} y =- (2-m)^{1/d} x -1 \end{aligned}$$

and, in particular (see Fig. 3), taking $m=m^\star$ we conclude that $T_d^{-1}(\widehat{\mathcal {D}}) \cap \{y\le 0\}$ is below

$$\begin{aligned} y = -(2-\widehat{m}^\star )^{1/d} x-1=-\left( \frac{3d-2}{d-1}\right) ^{1/d}x-1. \end{aligned}$$

(3.3)

The image $T_d(\widehat{{\mathcal {D}}})$. We notice that since $\{p_0,p_1\}$ is a two-cycle we have $T_d(p_0)=T_d^{-1}(p_0)=p_1,$ so that $T_d(\widehat{{\mathcal {D}}})$ is attached to $p_1$ as it was the case of $T_d^{-1}(\widehat{{\mathcal {D}}})$.

We denote by $\widehat{\gamma }_m(t)$ the image by $T_d$ of the segment $\{(t,1+mt)\mid t\in [0,1/(1-m)] \}$, with $m \in [\widehat{m}^\star ,m^-]$. Hence, $\widehat{\gamma }_m(t) = T_d(t,1+mt) =:(\widehat{x}_m(t),\widehat{y}_m(t))$ where

$$\begin{aligned} \widehat{x}_m(t) = mt +1 - ( (m+1)t +1)^d, \qquad \widehat{y}_m(t) = mt +1 - 2( (m+1)t +1)^d.\nonumber \\ \end{aligned}$$

(3.4)

To simplify notation, we write $x(t):= \widehat{x}_m(t)$ and $y(t):= \widehat{y}_m(t)$ and $\gamma (t)=\widehat{\gamma }_m(t)$ unless it is strictly necessary to show the dependence in m. The derivatives are given by

$$\begin{aligned} \begin{array}{ll} x^\prime (t) = m - d(m+1)( (m+1)t +1)^{d-1}, & y^\prime (t) = m - 2 d(m+1)( (m+1)t +1)^{d-1}, \\ x^{\prime \prime }(t) = - d(d-1)(m+1)^2( (m+1)t +1)^{d-2}, & y ^{\prime \prime }(t) = - 2 d(d-1)(m+1)^2( (m+1)t +1)^{d-2}. \end{array}\nonumber \\ \end{aligned}$$

Since $t\in [0,1/(1-m)]$ and $m<-1$, we have the inequalities

$$\begin{aligned} 0< \frac{2}{1-m} = \frac{m+1}{1-m} + 1 <(m+1)t +1 \le 1. \end{aligned}$$

Then, for $d\ge 3$ (odd), we have

$$\begin{aligned}{x}''(t)<0 \qquad \text {and} \qquad {y}''(t) <0.\end{aligned}$$

Next lemma provides basic estimates on the parametrization $\gamma (t)$.

Lemma 3.2

Let $m\in [\widehat{m}^\star , m^-]$ and $t\in [0,1/(1-m)]$. The following conditions hold.

• (a) $x(0)=0$, $x(\frac{1}{1-m})< \frac{1}{1-m}[1 -\frac{2}{\sqrt{e}}] <0$, $x(t)<0$ for $t\ne 0$, and $y(t)<0$.

• (b) $x' (t) \le 0$ with $x^{\prime }(t)=0$ if and only if $t=0$ and $m=\widehat{m}^{\star }$.

• (c) $y'(t) >0$ for $m=m^{\star }$.

Proof

The proof of the items follows from some computations based on the expressions of x(t), y(t) and their derivatives above.

Easily ${x}(0)=0$. On the one hand, we have

$$\begin{aligned} \begin{aligned} x\left( \frac{1}{1-m}\right) = \frac{1}{1-m} - \left( \frac{2}{1-m}\right) ^d&= \frac{1}{1-m}\left[ 1-2\left[ \left( 1+ \frac{1}{2(d-1)}\right) ^{2(d-1)}\right] ^{-1/2} \right] \\&< \frac{1}{1-m}\left[ 1 -\frac{2}{\sqrt{e}}\right] <0. \end{aligned} \end{aligned}$$

On the other hand, $x^{\prime }(0) = m-d(m+1)\le (-1-\frac{1}{d-1} ) (1-d) -d =0$ where the equality only holds for $m=\widehat{m}^{\star }$ and $x^{\prime \prime }(t)<0$ (see (3.1)). Hence, $x^{\prime }(t) <0$ (unless $t=0$ and $m=\widehat{m}^{\star }$ where $x_{m^\star }^{\prime }(0)=0$) and so x(t) is decreasing (and negative unless $t=0$). Finally, we have $y(t) = x(t) - ( (m+1)t +1)^d <0$. All together implies (a) and (b).

If $m=\widehat{m}^\star$, using (3.1) we have

$$\begin{aligned} {y}'_{m^{\star }}(t)&= m^{\star } - 2 d(m^{\star }+1)( (m^{\star }+1)t +1)^{d-1} \ge \frac{d}{d-1} \left[ -1+ 2\left( \frac{-1}{d-1} t + 1\right) ^{d-1}\right] \\&\ge \frac{d}{d-1} \left[ -1+ 2\left( 1 - \frac{1}{2d-1}\right) ^{d-1}\right] = \frac{d}{d-1} \left[ -1+ 2\left( 1+ \frac{1}{2(d-1)}\right) ^{1-d}\right] \\ &> \frac{d}{d-1} \left[ -1+ \frac{2}{\sqrt{e}}\right] >0 \end{aligned}$$

that proves (c). $\square$

Since $x'(t)<0$, the function x(t) is invertible. If $t=t(x)$ is the inverse map of x(t), then (the image of) $\gamma (t)$ can be represented as the graph of the function $h(x):=h_m(x):= y\circ t(x)$. From its definition, the function h is smooth.

Lemma 3.3

We have that $h(x)=y \circ t (x)$ is concave.

Proof

Taking derivatives we have

$$\begin{aligned} h^{\prime }(x) = \frac{y^{\prime }}{x^{\prime }}\circ t(x)\qquad \text{ and } \qquad h^{\prime \prime }(x) =\frac{1}{(x^{\prime })^2}\left[ y^{\prime \prime } - x^{\prime \prime } \frac{y^{\prime }}{x^{\prime }} \right] \circ t(x). \end{aligned}$$

From the expressions of x and y and their derivatives (see (3.4) and the derivatives below), we have ${y}^{\prime }(t) = {x}^{\prime }(t)- d(m+1)( (m+1)t +1)^{d-1}$ and ${y}^{\prime \prime }(t) = 2 {x}^{\prime \prime }(t)$. Therefore

$$\begin{aligned} h^{\prime \prime }(x) =\frac{x^{\prime \prime }}{(x^{\prime })^2}\left[ 2 - \frac{y^{\prime }}{x^{\prime }} \right] \circ t(x) \quad \text{ and } \quad 2-\frac{y'}{x'} = 1+ \frac{d(m+1)( (m+1)t +1)^{d-1}}{x'} >1, \end{aligned}$$

concluding $h^{\prime \prime }(x) <0$ and hence h is concave (remember that the result is valid for all values of m in the range). $\square$

Now we fix $m=\widehat{m}^\star$. We claim that (the image of) $\widehat{\gamma }_{\widehat{m}^\star }$ belongs to $\{x\le 0,y\le 0 \}$ and it is above $T^{-1}_d(\widehat{\mathcal {D}})$. To check the claim, accordingly to the previous study of $T^{-1}_d(\widehat{\mathcal {D}})$ it is sufficient to check that $\widehat{\gamma }_{\widehat{m}^\star }$ is above the line $y= -(2-\widehat{m}^\star )^{1/d} x-1$ introduced in (3.3). Moreover, since (the image of) $\widehat{\gamma }_{\widehat{m}^\star }$ is the graph of a concave function it is enough to check that

$$\begin{aligned} y_{\widehat{m}^{\star }}\left( \frac{1}{1-\widehat{m}^{\star }}\right) > -(2-\widehat{m}^{\star })^{1/d} \ x_{\widehat{m}^{\star }}\left( \frac{1}{1-\widehat{m}^{\star }}\right) -1. \end{aligned}$$

This is equivalent to

$$\begin{aligned} \frac{1}{1-\widehat{m}^\star } - 2 \left( \frac{2}{1-\widehat{m}^\star }\right) ^d + (2-\widehat{m}^\star )^{1/d} \left[ \frac{1}{1-\widehat{m}^\star } - \left( \frac{2}{1-\widehat{m}^\star }\right) ^d \right] +1>0. \end{aligned}$$

If we substitute $\widehat{m}^\star = -1-\frac{1}{d-1}$, the above inequality writes as

$$\begin{aligned} & 1 - \left( \frac{2}{2+1/(d-1)}\right) ^d + \left[ \frac{1}{2+1/(d-1)} - \left( \frac{2}{2+1/(d-1)}\right) ^d \right] \nonumber \\ & \quad \times \left[ 1+\left( 3+ \frac{1}{d-1}\right) ^{1/d} \right] >0. \end{aligned}$$

(3.5)

On the one hand we have that

$$\begin{aligned} \left( \frac{2}{2+1/(d-1)}\right) ^d < \left( \frac{1}{1+1/(2d)}\right) ^d \le \left( \frac{6}{7} \right) ^3. \end{aligned}$$

On the other hand we have that

$$\begin{aligned}&\frac{1}{2+1/(d-1)} - \left( \frac{2}{2+1/(d-1)}\right) ^d\\&\quad = \frac{1}{2+1/(d-1)} \left[ 1-2 \left[ \left( 1+\frac{1}{2(d-1)} \right) ^{2(d-1)} \right] ^{-1/2} \right] \\&\quad \ge \frac{1}{2+1/(d-1)} \left[ 1-2\left( 1+\frac{1}{4}\right) ^{-2}\right] \ge \frac{-7}{50}, \end{aligned}$$

and

$$\begin{aligned} 0<1+\left( 3+ \frac{1}{d-1}\right) ^{1/d} < 1+\left( 3+ \frac{1}{2}\right) ^{1/3} = 1+ (7/2)^{1/3}. \end{aligned}$$

Hence, to prove (3.5) it is enough to check that

$$\begin{aligned} 1-\left( \frac{6}{7}\right) ^3-\frac{7}{50}\left( 1+\left( \frac{7}{2}\right) ^{1/3}\right) \approx 0.02 >0. \end{aligned}$$

Moreover, we also claim that $\widehat{\gamma }_{m^-}$ is below $T^{-1}_d(\widehat{\mathcal {D}})$. This easily follows from the fact that, by the description of the preimage $T^{-1}_d(\widehat{\mathcal {D}})$, the left boundary of $T^{-1}_d(\widehat{\mathcal {D}})$ is the graph of a convex function $x=g(y)$ while $\widehat{\gamma }_{m^-}$ is the graph of a concave function $y=h_{m^-}(x)$ and both graphs are tangent at $p_1$.

It follows from lemmas above that we have a deep control on the left and right boundaries of $T_d(\widehat{\mathcal {D}})$, and their relative position with respect to the set $T_d^{-1}(\widehat{\mathcal {D}})$. See the right picture of Fig. 3. Now we close the argument by controlling the image of $\partial \widehat{\mathcal {D}}\cap \{y=x\}$.

Lemma 3.4

The upper piece of the boundary of $T_d(\widehat{\mathcal {D}})$ is the image by $T_d$ of the piece of the boundary $\left\{ (t,t) \mid \frac{1}{1-\widehat{m}^\star } \le t \le \frac{1}{1-m^-} \right\}$ of $\widehat{\mathcal {D}}$. It can be represented as the graph of an increasing function and is contained in $\{ x<0, \, y<0\}$.

Proof

We introduce

$$\begin{aligned} T_d(t,t) = (t-(2t)^d, t-2(2t)^d )=: (\xi (t), \eta (t)),\quad (1-\widehat{m}^\star )^{-1} \le t \le (1-m^-)^{-1}. \end{aligned}$$

Taking first and second derivatives, we have

$$\begin{aligned} \xi '(t)= & 1-2d (2t)^{d-1}, \quad \eta '(t) = 1-4d (2t)^{d-1},\\ \xi ''(t)= & -4d(d-1) (2t)^{d-2}\, \textrm{and}\, \eta ''(t) = 2 \xi ''(t). \end{aligned}$$

First we check that, in the corresponding domain, $\xi '(t)<0$ and $\eta '(t)<0$. This follows from $\xi ''(t)<0, \ \eta ''(t)<0$ and

$$\begin{aligned} \xi '\left( \frac{1}{1-\widehat{m}^\star }\right)&= 1-2d \left( \frac{2}{1-\widehat{m}^\star }\right) ^{d-1} = 1-2d\frac{1}{\left( 1+ \frac{1}{2(d-1)}\right) ^{d-1}}<1-\frac{2d}{\sqrt{e}} <0, \end{aligned}$$

and

$$\begin{aligned} \eta '\left( \frac{1}{1-\widehat{m}^\star }\right) = \xi '\left( \frac{1}{1-\widehat{m}^\star }\right) - 2d \left( \frac{2}{1-\widehat{m}^\star }\right) ^{d-1} <0. \end{aligned}$$

The condition $\xi '(t)<0$ implies that $\xi (t)$ is invertible. Let $t=t(\xi )$ be its inverse function and $\eta = f(\xi ):= \eta \circ t(\xi )$. The curve $T_d(t,t)$ is the graph of f and

$$\begin{aligned} f^{\prime }= \frac{\eta ^{\prime }}{\xi ^{\prime }} \circ t(x)>0. \end{aligned}$$

Moreover, since

$$\begin{aligned} \xi \left( \frac{1}{1-\widehat{m}^\star }\right)&= \frac{1}{1-\widehat{m}^\star }\left( 1-2\left( \frac{2}{1-\widehat{m}^\star }\right) ^{d-1}\right) \!=\! \frac{1}{1-\widehat{m}^\star }\left( 1-2\frac{1}{(1\!+\! \frac{1}{2(d-1)})^{d-1}}\right) \\&< \frac{1}{1-\widehat{m}^\star }\left( 1-\frac{2}{\sqrt{e}} \right) <0, \end{aligned}$$

we have $\xi (t)<0$ and $\eta (t) = \xi (t) -(2t)^d <0$. $\square$

Up to this point, we have completed the study of the geometry and relative positions of $T_d(\widehat{\mathcal {D}})$ and $T_d^{-1}(\widehat{\mathcal {D}})$ (see the right picture of Fig. 3). Next two lemmas show that there is a piece of $W_{p_1}^s$ attached to $p_1$, being tangent to $y= m^{-} x-1$ at $p_1$, included in $T_d^{-1}(\widehat{\mathcal {D}})$ and connecting $p_1$ with a point in $\partial T_d^{-1}(\widehat{\mathcal {D}}) \cap \{y=-x\}, \ x<0$.

Let $(x_0,y_0) \in T_d^{-1}(\widehat{\mathcal {D}})$. Then, we write $(x_{2k},y_{2k}):=T_d^{2k}(x_0,y_0)$. The first lemma characterize the dynamics of points in $T_d^{-1}(\widehat{\mathcal {D}})$ whose all iterates under $T_d^{2}$ remain in $T_d^{-1}(\widehat{\mathcal {D}})$.

Lemma 3.5

If $(x_0,y_0) \in T_d^{-1}(\widehat{\mathcal {D}})$ and $(x_{2k},y_{2k}) \in T_d^{-1}(\widehat{\mathcal {D}})$ for all $k\ge 0$ then we have that $(x_{2k},y_{2k}) \rightarrow p_1=(0,-1)$ as $k \rightarrow \infty$.

Proof

First we note that $y_2<0$. Indeed, $(x_{2},y_{2}) \in T_d (\widehat{\mathcal {D}})$ and by Lemma 3.4

$$\begin{aligned} \sup \left\{ \eta (t) \mid \ \frac{1}{1-\widehat{m}^\star } \le t \le \frac{1}{1- m^-}\right\} = \eta \left( \frac{1}{1-\widehat{m}^\star }\right) <0. \end{aligned}$$

Moreover, the right boundary of $T_d (\widehat{\mathcal {D}})$ is given by $\widehat{\gamma }_{\widehat{m}^\star }(t)= (\widehat{x}_m(t),\widehat{y}_m(t))$ and, by Lemma 3.2(a), $\widehat{y}_{\widehat{m}^\star } (t)< \widehat{y}_{\widehat{m}^\star } (\frac{1}{1-\widehat{m}^\star }) = \eta (\frac{1}{1-\widehat{m}^\star })<0$.

Now, let $(x_0,y_0)$ as in the statement with $y_0<0$. Using that $T_d^{-1}(\widehat{\mathcal {D}}) \cap \{y\le 0 \}$ is below the line $y =-(2-\widehat{m}^\star )^{1/d} x -1$ we have that

$$\begin{aligned} x_0< \frac{y_0+1}{-(2-\widehat{m}^\star )^{1/d}}. \end{aligned}$$

(3.6)

First, we compute

$$\begin{aligned} (x_1,y_1) = T_d (x_0,y_0)=(y_0-(x_0+y_0)^d,y_0-2(x_0+y_0)^d ). \end{aligned}$$

We observe that $(x_1,y_1)$ belongs to the line $y=2x-y_0$.

By the definition of $\widehat{\mathcal {D}}$, we have that $x_1$ is less than the first coordinate of the intersection $\{ y=2x-y_0\} \cap \{ y= m^{-} x +1\}$, i.e. $x_1 < \frac{1+y_0}{2- m^-}$. Moreover, using (3.6),

$$\begin{aligned} 0\le x_1 < \frac{-(2-\widehat{m}^\star )^{1/d}}{2- m^-} x_0. \end{aligned}$$

Next we bound

$$\begin{aligned} \left| \frac{-(2-\widehat{m}^\star )^{1/d}}{2- m^-}\right| < \frac{(3+\frac{1}{d-1})^{1/d}}{3} \le \frac{1}{3} \left( \frac{7}{2}\right) ^{1/3}. \end{aligned}$$

Now we deal with the next iterate $(x_2,y_2) =(y_1- (x_1+y_1)^d,y_1-2(x_1+y_1)^d )$. Since $(x_1,y_1) \in \widehat{\mathcal {D}}$, $0< x_1+y_1 \le 1$ and $y_1 \ge \widehat{m}^\star x_1 +1$ we conclude that

$$\begin{aligned} 0\ge x_2 = y_1-(x_1+y_1)^d \ge y_1 -1 \ge \widehat{m}^\star x_1. \end{aligned}$$

Consequently,

$$\begin{aligned} |x_2| \le |\widehat{m}^\star | x_1 \le \frac{3}{2} \frac{1}{3} \left( \frac{7}{2}\right) ^{1/3}|x_0| \le \frac{4}{5} |x_0|. \end{aligned}$$

Recursively, we obtain that $|x_{2k}| \le (\frac{4}{5} )^k |x_0|$ and this implies $x_{2k} \rightarrow 0$, Since, by hypothesis, $(x_{2k},y_{2k}) \in T_d^{-1}(\widehat{\mathcal {D}})$ for all $k \ge 0$ we conclude that $y_{2k} \rightarrow -1$. $\square$

Lemma 3.6

The set $T^{-1}_d(\widehat{\mathcal {D}})$ contains a piece of $W^s_{p_1}$ joining the point $p_1$ with a point in $T^{-1}_d(\widehat{\mathcal {D}}) \cap \{y=-x\}$.

Proof

We will use the same argument we have used in Fontich et al. (2024). Take $I_0$ any segment joining the right and left boundaries of $T^{-1}_d(\widehat{\mathcal {D}})$. By the previous lemmas, $T^2_d(I_0)$ is a curve contained in $T _d(\widehat{\mathcal {D}})$ joining its right and left boundaries which are outside $T^{-1}_d(\widehat{\mathcal {D}})$, thus it has to cross the right and left boundaries of $T^{-1}_d(\widehat{\mathcal {D}})$.

We define $I_1 = T^{-2}_d (T^{2}_d (I_0) \cap T^{-1}_d(\widehat{\mathcal {D}})) \subset I_0$ and, in general,

$$\begin{aligned} I_n = T^{-2n}_d (T^{2n}_d (I_{n-1}) \cap T^{-1}_d(\widehat{\mathcal {D}})) \subset I_{n-1}, \qquad n\ge 1. \end{aligned}$$

Then, $\{I_n\}_{n\ge 1}$ is a sequence of nested compact sets and $I_\infty := \bigcap _{n\ge 1}I_n \ne \emptyset$. This set has the property that all points in $I_\infty$ are such that all their iterates stay in $T^{-1}_d(\widehat{\mathcal {D}})$ and, by Lemma 3.5, converge to $p_1$. Therefore, $I_\infty = W^s_{p_1} \cap T^{-1}_d(\widehat{\mathcal {D}})\cap I_0$. $\square$

Proof of Proposition 3.1

We will see that the above description of the relative positions of $T_d({\mathcal {D}})$ and $T_d^{-1}(\widehat{{\mathcal {D}}})$ (neighbourhoods of pieces of $W^u_{p_0}$ and $W^s_{p_1}$, respectively) implies a heteroclinic intersection between the stable manifold of $p_1$ and the unstable manifold of $p_0$. Unless it is necessary, we drop the dependence on the parameter m.

On the one hand, in Fontich et al. (2024, Lemma 5.4) it is proven that there is a connected piece of $W^u_{p_0}$ contained in $T_d({\mathcal {D}})$ joining $p_0$ with some point in $\partial T_d({\mathcal {D}}) \cap \{y=x\}$. On the other hand, the above lemmas show that there is a piece of $W^s_{p_1}$ contained in $T_d^{-1}(\widehat{\mathcal {D}})$ which joints $p_1=(0,-1)$ with a point in $T_d^{-1}(\widehat{\mathcal {D}})\cap \{y=-x\}$.

We claim that the line L given by $\{y= m^-x-1\}$, tangent to the left boundary of $T_d^{-1}(\widehat{\mathcal {D}})$ at $(0,-1)$, intersects in two points the right boundary of $T_d(D)$ which is given by the curve $\gamma _{m}(t)$ in (2.5) with $m=m^\star =7/2$. If we write $\gamma _{m^\star }(t)=\left( X(t),Y(t)\right)$ we have

$$\begin{aligned} {X(t)}= & m^\star t -1 - ( (m^\star +1)t -1)^d, \\ {Y(t)}= & m^\star t -1 - 2( (m^\star +1)t -1)^d, \quad t\in \left[ 0, \frac{1}{m^\star +1}\right] . \end{aligned}$$

See Fig. 4. To check the claim, recall that $\frac{-6}{\sqrt{13}+1} \le m^-< -1$. We consider the auxiliary function

$$\begin{aligned} \phi (t)&= Y(t) - m^- X(t) +1 \\&=(2- m^-) [(1- (m^\star +1)t )^d + m^\star t -1] -m^\star t +2 , \qquad t\in \left[ 0, \frac{1}{m^\star +1}\right] , \end{aligned}$$

which measures whether $\gamma _{m^\star }(t)$ is below, above or on the line L. We have

$$\begin{aligned} \begin{aligned}&\phi (0)= 2>0, \qquad \phi ( \frac{1 }{m^\star +1})= \frac{m^- +m^\star }{m^\star +1}>, \qquad \text { and} \\&\phi ''(t) = (2- m^- ) d(d-1)(m^\star +1)^2 (1- (m^\star +1)t)^{d-2}>0, \qquad t\in \left[ 0, \frac{1}{m^\star +1}\right] . \end{aligned} \end{aligned}$$

Accordingly, in order to see that $\phi$ has two zeros in its domain it is enough to show that there is a point $t_1$ in $(0, \frac{1}{1+m^\star })$ such that $\phi (t_1)<0$. We take $t_1=1/8$ and, using that $m^- > - 4/3$, we have

$$\begin{aligned}\phi (1/8)=(2- m^-) \left( \left( \frac{7}{16} \right) ^{d} -\frac{9}{16}\right) + \frac{25}{16}< \frac{10}{3} \left( \frac{7^3}{16^3}-\frac{9}{16}\right) +\frac{25}{16}<0. \end{aligned}$$

Therefore, $W^u_{p_0}$ has to cross L.

Next we claim that $T_d^{-1}(\widehat{\mathcal {D}}) \cap \{y=0\}$ is a segment $[a_-, a_+] \times \{0\}$ with $a_+ < -3/5$. To see this claim we look for the intersection of the right and left boundaries of $T_d^{-1}(\widehat{D})$, given by $\Gamma _{\widehat{m}^\star }(t) =( \widehat{\alpha }_{\widehat{m}^\star }(t), \widehat{\beta }_{\widehat{m}^\star }(t) )$ and $\Gamma _{m^-}(t) =( \widehat{\alpha }_{m^-}(t), \widehat{\beta }_{m^-}(t) )$, respectively, with $\{y=0\}$. We recall that $\widehat{m}^\star =-1-1/(d-1)$ and

$$\begin{aligned} \widehat{\alpha }_{\widehat{m}^\star }(t)= & ({\widehat{m}^\star } -2) t +1 + ( (1- {\widehat{m}^\star })t -1)^{1/d}, \\ \widehat{\beta }_{\widehat{m}^\star }(t)= & (2-{\widehat{m}^\star }) t -1, \qquad t\in \left[ 0, \frac{1}{1-\widehat{m}^\star }\right] . \end{aligned}$$

The value $t=t_2$ such that $\widehat{\beta }_{\widehat{m}^\star }(t) =0$ is $t_2= \frac{1}{2-\widehat{m}^\star } \in \left[ 0, \frac{1}{1-\widehat{m}^\star }\right]$, and

$$\begin{aligned} a_+= \widehat{\alpha }_{\widehat{m}^\star }(t_2) =- \left( \frac{1}{2-\widehat{m}^\star }\right) ^{1/d} = - \left( \frac{1}{3 + 1/(d-1) }\right) ^{1/d} \le -\left( \frac{2}{7}\right) ^{1/3} < -\frac{3}{5}. \end{aligned}$$

In the same way, denoting $t_3$ the value such that $\widehat{\beta }_{m^-}(t_3)=0$, we obtain $a_-= - \left( 2-\widehat{m}^\star \right) ^{-1/d} < a_+$

Putting together the information of the two previous claims we get that when $y=0$, $W^u_{p_0}$ is to the right of $W^s_{p_1}$ and that there exists some $y = y^0<0$ for which $\gamma _{m^{\star }}$ is to the left of L and therefore $W^u_{p_0}$ has to be at the left of $W^s_{p_1}$. This finish the proof of the proposition. See Fig. 4. $\square$

Fig. 4

Sketch of the arguments providing the (topological, not necessarily transversal as it is shown in the picture) intersection of the stable and unstable manifold of the hyperbolic two-cycle $\{p_0,p_1\}$ (Theorem A). The green dots indicates the two intersections between $\gamma _{m^\star }(t)$ and the line $L:=\{y= m^{-}x-1\}$ (Color figure online)

Proof of Theorem A

Since d is odd, the map $T_d$ is symmetric with respect to $(x,y)\mapsto (-x,-y)$. Proposition 3.1 provides a (maybe non-transversal) heteroclinic point q in $T_d^{-1}(\widehat{\mathcal {D}}) \cap T_d({\mathcal {D}})$. In any case at this point the manifolds cross each other. Therefore $\overline{q}=-q$ is also a heteroclinic point. By symmetry, at the point $\overline{q}$ the unstable manifold of $p_1$ intersects the stable manifold of $p_0$. We know that the unstable manifold is analytic. The stable manifold is analytic in a neighbourhood of q since the globalization of the local manifold has not meet $\{y=x\}$ yet. Since the manifolds do not coincide, they have a finite order contact.

Since we do not know if the intersection is transversal, we cannot apply the $\lambda$-Lemma of Palis in Palis (1969). However, we can apply the singular $\lambda$-Lemma in Rayskin (2003). In the two dimensional case, it asserts that the iteration of a disc in the unstable manifold accumulates in a $C^1$ manner to the unstable manifold of $p_0$, except for an arbitrarily small neighbourhood of $p_0$.

Now consider a piece of the connected component of the unstable manifold of $p_0$ in $T_d^{-1}(\widehat{\mathcal {D}}) \cap T({\mathcal {D}})$ joining two points of the upper and lower boundaries of $T^{-1}(\widehat{\mathcal {D}})$, respectively. Then, by the singular $\lambda$-Lemma, the unstable manifold of $p_1$ will have discs arbitrary $C^1$-close to the unstable manifold of $p_0$ and therefore the discs will be in $T_d^{-1}(\widehat{\mathcal {D}}) \cap T_d ({\mathcal {D}})$.

Finally, using the same argument as in the end of the proof of the first part of Theorem A these discs should have an intersection with the stable manifold of $p_1$, thus providing the desired homoclinic point. $\square$

4 Proof of Theorem B

In the previous section, we have proven the existence of homoclinic points associated with the stable and unstable of the cycle $\{p_0,p_1\}$. Using this fact, in this section we demonstrate that stable and the unstable manifolds of $p_1$ intersect in a transverse homoclinic point. Our approach is inspired in the work of Churchill and Rod (1980). However, there is an important difference. In Churchill and Rod (1980) the authors deal with analytic area preserving maps and can use tools as the Birkhoff normal form, while our map is not area preserving and it is not an analytic diffeomorphism. Our presentation uses the special structure of the map and the fact that we can linearize the map $T^2_d$ around $p_1$ which a $C^\infty$ conjugation.

For the point $p_1=(0,-1)$, we will denote by

$$\begin{aligned} W_{\textrm{loc}}^s:=W_{\textrm{loc},p_1}^s, \qquad W_{\textrm{loc}}^u:=W_{\textrm{loc},p_1}^u, \qquad W^s:=W_{p_1}^s \qquad \text { and} \qquad W^u:=W_{p_1}^u \end{aligned}$$

the local stable, local unstable, global stable and global unstable manifolds associated with $p_1$ for the map $T_d^2$, respectively. The size of the local manifolds will be as small as we need.

We split the proof of Theorem B into several lemmas. Given $z\in \mathbb {R}^2$ we let $B_\varepsilon (z)$ be the open ball centred at z and radius $\varepsilon >0$.

Lemma 4.1

Let $\varepsilon >0$ be small enough. Then, there exist two points $q_s$ and $q_u$ in $B_\varepsilon (p_1)$ such that

$$\begin{aligned} q_s \in W_{\textrm{loc}}^s \cap W^u \qquad \text {and} \qquad q_u \in W_{\textrm{loc}}^u \cap W^s. \end{aligned}$$

Moreover, there exist analytic local parametrizations of $W^s$ around $q_u$ and of $W^u$ around $q_s$ given by $\{\phi ^s(u)\mid \, u\in (-\delta , \delta )\}$ with $\phi ^s(0)=q_u$ and $\{\phi ^u(u)\mid \, u\in (-\delta , \delta )\}$ with $\phi ^u(0)=q_s$ for some $\delta >0$ small.

Since the manifolds do not coincide, the above intersections (at the points $q_s$ and $q_u$) have finite order contact.

Fix $\varepsilon _1>0$ small enough such that $B_{\varepsilon _1}(q_s)\subset B_{\varepsilon }(p_1)$ and $B_{\varepsilon _1}(q_u)\subset B_{\varepsilon }(p_1)$. We denote by $\widehat{W}^u$ the piece of $W^u \subset B_{\varepsilon _1}(q_s)$ and by $\widetilde{W}^s$ the piece of $W^s \subset B_{\varepsilon _1}(q_u)$.

Proof

Fix $\varepsilon >0$ small enough and consider local manifolds $W_{\textrm{loc}}^s, \ W_{\textrm{loc}}^u$ contained in $B_\varepsilon (p_1)$. Let $q\in W^s \cap W^u$ be the point determined by the topological transversal intersection of the stable and the unstable manifolds of $p_1$ for the map $T_d^2$ given by Theorem A. By iterating forward this point by $T_d^2$ and $(T_d^2)^{-1}$ we obtain the existence of $q_s$ and $q_u$ in $B_\varepsilon (p_1)$, respectively. Moreover, since $W_{\textrm{loc}}^s$ and $W^u$ are analytic we have that $W_{\textrm{loc}}^s \cap W^u$ intersect with finite order contact (otherwise they would coincide). Then, there exists $\phi ^u$ as claimed. By construction, there exists $n_0>0$ such that

$$\begin{aligned} T_d^{-n_0}(q_s)=q_u. \end{aligned}$$

According to the previous arguments if

$$\begin{aligned} T_d^{-j}(q_s) \cap \{y=x\} = \emptyset , \qquad j=1,\ldots , n_0-1, \end{aligned}$$

(4.1)

then $W_{\textrm{loc}}^u \cap W^s$ intersect at $q_u$, $W^s$ is analytic in a neighbourhood of $q_u$ and the intersection has a finite order contact and the lemma follows. Now, we consider the case that there exists a finite sequence of natural numbers $0< j_1< j_2< \cdots< j_\ell <n_0$, $1\le \ell < n_0$, such that

$$\begin{aligned} T_d^{-j_k}(q_s) \cap \{y=x\}=:\textbf{r}_k \in {\mathbb {R}}^2, \qquad k=1,\ldots , \ell . \end{aligned}$$

(4.2)

Note that $\textbf{r}_k=(r_k,r_k), \ r_k \in {\mathbb {R}}$, and hence, $T_d^{-1}(r_k,r_k)=(-r_k,r_k)$. First, we deal with $\textbf{r}_1$, the first time the globalization of $W^s_\textrm{loc}$ meets $\{y=x\}$ so that $W^s$ is analytic from $p_1$ to this point. Thus, near $\textbf{r}_1$ the stable manifold $W^s$, is analytic and can be parametrized as

$$\begin{aligned} \phi (t)=\left( r_1+t^{\alpha _1}\left( a_1+f_1(t)\right) ,r_1+t^{\beta _1}\left( b_1+g_1(t)\right) \right) , \qquad |t| < \delta _1, \end{aligned}$$

where $\alpha _1,\beta _1 \in {\mathbb {N}}$, $a_1,b_1 \in {\mathbb {R}}{\setminus } \{0\}$, $f_1(t),g_1(t)$ are analytic, satisfy $f_1(0)= 0$ and $g_1(0)= 0$ and $\delta _1>0$ is small enough. Since $W^s \not \subset \{y=x\}$ we have

$$\begin{aligned} t^{\alpha _1}\left( a_1+f_1(t)\right) - t^{\beta _1}\left( b_1+g_1(t)\right) \not \equiv 0. \end{aligned}$$

Using the expression of $T_d^{-1}$ (see Eq. 2.1) we have

$$\begin{aligned} T_d^{-1}\left( \phi (t)\right) =\left( \begin{array}{l} -r_1-2t^{\alpha _1}\left( a_1+f_1(t)\right) +t^{\beta _1}\left( b_1+g_1(t)\right) +t^{\gamma _1/d}\left( 1+O(t)\right) ^{1/d}\\ \ \ r_1+2t^{\alpha _1}\left( a_1+f_1(t)\right) -t^{\beta _1}\left( b_1+g_1(t)\right) \end{array}\right) , \end{aligned}$$

where $\gamma _1\ge \min \{\alpha _1,\beta _1\}$. Since d is odd we can reparametrize the curve $\phi (t)$ using the new parameter $u=t^{1/d}$ to obtain $\widehat{\phi }(u) =\phi (u^{d})$ analytic and

$$\begin{aligned} T_d^{-1}\left( \widehat{\phi }(u)\right) = T_d^{-1}\left( \phi (u^d)\right) =\left( \begin{array}{l} -r_1+O(u^{\widehat{\alpha }_1})\\ \ \ r_1+O(u^{\widehat{\beta }_1}) \end{array}\right) , \end{aligned}$$

with $\widehat{\alpha }_1$, $\widehat{\beta }_1 \in \mathbb {N}$.

We conclude thus that $W^s$ admits an analytic parametrization in a sufficiently small neighbourhood of $T_d^{-1}(r_1,r_1)=(-r_1,r_1)$. Repeating the same procedure a finite number of times it is clear that $W^s$ intersects $W_{\textrm{loc}}^u$ with finite order contact at the point $q_u$. $\square$

The translation

$$\begin{aligned} {\mathcal {T}}: (\widehat{x},\widehat{y})\mapsto (x,y)=(\widehat{x},\widehat{y}-1) \end{aligned}$$

moves $p_1$ to the origin. For simplicity, we write the new coordinates again as (x, y). Observe that (in the new coordinates) $T_d^2(0,0)=(0,0)$ and that

$$\begin{aligned} DT_d^2(0,0)=\left( \begin{array}{cc} 3d^2-2d & \ \ 3d^2-4d+1 \\ 6d^2-2d & \ \ 6d^2-6d+1 \end{array}\right) . \end{aligned}$$

(4.3)

The eigenvalues and eigenvectors are given in (2.3) and (2.4), respectively.

We will denote

$$\begin{aligned} \lambda :=\lambda _d^{+}> 1,\qquad \mu :=\lambda _d^{-} < 1, \qquad m_{\lambda }:= m^{+}_d \qquad \text { and} \qquad m_{\mu }:= m^{-}_d \end{aligned}$$

(4.4)

(we drop the dependence on d unless it is strictly necessary). We recall from Sect. 2 that

$$\begin{aligned} \lambda > 57, \quad 1/9<\mu<0.1556, \quad 2<m_\lambda<2.3028 \quad \text {and} \quad -1.3027<m_\mu <-1. \end{aligned}$$

(4.5)

We parametrize the local stable and unstable manifolds associated with the origin by the x-variable so that the expressions can be written as $y=\Psi ^s(x)$ and $y=\Psi ^u(x)$, respectively. We obviously have

$$\begin{aligned} \frac{\text {d} \Psi ^s}{\text {d} x}(x)|_{x=0} =m_\mu \qquad \text { and} \qquad \frac{\text {d} \Psi ^u}{\text {d} x}(x)|_{x=0} = m_\lambda . \end{aligned}$$

(4.6)

Next step is to introduce local analytic coordinates $( \widehat{\xi },\widehat{\eta })$ around (0, 0) so that the expression of the local stable and unstable manifolds would be $\widehat{\eta }=0$ and $\widehat{\xi }=0$, respectively.

Lemma 4.2

We consider the local change of variables

$$\begin{aligned} (x,y)\mapsto (\widehat{\xi },\widehat{\eta })=\Theta (x,y):=( y-\Psi ^u(x), y-\Psi ^s(x)). \end{aligned}$$

Then, for $\varepsilon >0$ small enough the local expression of $T_d^2$ in $B_\varepsilon (0,0)$ is given by

$$\begin{aligned} {\mathcal {F}}(\widehat{\xi },\widehat{\eta })={\mathcal {L}}(\widehat{\xi },\widehat{\eta }) + {\mathcal {N}}(\widehat{\xi },\widehat{\eta }), \end{aligned}$$

where

$$\begin{aligned} {\mathcal {L}}(\widehat{\xi },\widehat{\eta })=(\lambda \widehat{\xi }, \mu \widehat{\eta }),\quad {\mathcal {N}}(0,0)=(0,0) \quad \text { and} \quad D{\mathcal {N}}(0,0) = 0. \end{aligned}$$

Moreover, the local change of coordinates, $(\widehat{\xi },\widehat{\eta })=\Theta (x,y)$, is analytic.

Proof

We claim that the new variables define a local change of coordinates around the origin. Indeed, since $\Psi ^s(x)$ and $\Psi ^u(x)$ are analytic, by the inverse function theorem we only need to check that

$$\begin{aligned} D\Theta (0,0) = \left( \begin{array}{cc} - \frac{\partial \Psi ^u}{\partial x}(x)|_{x=0} & 1 \\ & \\ - \frac{\partial \Psi ^s}{\partial x} (x)|_{x=0} & 1 \end{array}\right) = \left( \begin{array}{cc} - m_\mu & 1 \\ & \\ - m_\lambda & 1 \end{array}\right) \end{aligned}$$

is non-singular and this is a direct consequence of (4.5). Clearly $\Theta$ is a local analytic diffeomorphism. See Fig. 5. $\square$

Fig. 5

The changes of coordinates corresponding to Lemma 4.2 and Lemma 4.4. In fact $\Theta$ in this figure includes a primer change of coordinates to move $p_1$ to the origin (Color figure online)

We will see next that ${\mathcal {F}}$ is ${\mathcal {C}}^{\infty }$-conjugate to its linear part ${\mathcal {L}}$. For this, we will apply Sternberg’s Theorem Sternberg (1958, Theorem 1). The following lemma checks a key hypothesis of that theorem.

Lemma 4.3

The eigenvalues $\lambda = \lambda _d^{+}>1$ and $\mu =\lambda _d^{-}<1$ of the linear part ${\mathcal {L}}$ of ${\mathcal {F}}$ at (0, 0) given in (4.4) and (2.3) are non-resonant.

Proof

We first note that $\lambda \mu =d^2$ (the determinant of $DT^2_d(0,0)$). It is easy to check by induction that

$$\begin{aligned} \lambda ^k=a_k+b_k\sqrt{\Delta } \qquad \text { and} \qquad \mu ^k=a_k-b_k\sqrt{\Delta }, \qquad k\ge 1, \end{aligned}$$

and

$$\begin{aligned} \lambda ^{-k}=\frac{1}{g^k} \left( a'_k-b'_k\sqrt{\Delta }\right) \qquad \text { and} \qquad \mu ^{-k} =\frac{1}{g^k} \left( a'_k+b'_k\sqrt{\Delta }\right) , \qquad k\ge 1, \end{aligned}$$

where $a_k,b_k, a'_k, b'_k \in {\mathbb {N}}$, $g=12 d^2 (6 d^2 - 4 d + 1)\in \mathbb {N}$ and $\Delta =9d^2-10d+1$ is not a perfect square. This means that $\lambda ^k, \mu ^k \in \mathbb {R}{\setminus } \mathbb {Q}$ for all $k\ne 0$.

There are two possible types of resonances:

$$\begin{aligned} \lambda =\lambda ^{n}\mu ^{m} \qquad \text { with} \qquad n,m\ge 0 \qquad \text{ and } \quad n+m\ge 2, \end{aligned}$$

(4.7)

and

$$\begin{aligned} \mu =\lambda ^{n}\mu ^{m} \qquad \text { with} \qquad n,m\ge 0 \qquad \text{ and } \quad n+m\ge 2. \end{aligned}$$

(4.8)

We deal with (4.7). We rewrite it as

$$\begin{aligned} 1= \lambda ^{n-m-1}(\lambda \mu )^{m} =\lambda ^{n-m-1} d^{2m}. \end{aligned}$$

(4.9)

We distinguish two cases: (a) $n\ne m+1$ and (b) $n= m+1$.

In case (a), since $\lambda ^{n-m-1} \in \mathbb {R}{\setminus } \mathbb {Q}$ and $d\in \mathbb {N}$ the previous equality is impossible. In case (b), m cannot be 0. Then, $(\lambda \mu )^{m}= d^{2m} \ge 9$ so that (4.9) is also impossible.

Concerning resonances of the form (4.8) the argument is completely analogous. $\square$

Theorem 1 in Sternberg (1958) provides a ${\mathcal {C}}^{\infty }$ local change of coordinates conjugating ${\mathcal {F}}$ to its linear part ${\mathcal {L}}$. From it we will obtain a near the identity conjugation.

Lemma 4.4

There is a conjugacy from ${\mathcal {F}}$ to its linear part ${\mathcal {L}}$ at the origin of the form

$$\begin{aligned} (\xi ,\eta ):=\Phi (\widehat{\xi },\widehat{\eta })=\left( \begin{array}{c} \widehat{\xi } (1+\phi _1(\widehat{\xi },\widehat{\eta })) \\ \widehat{\eta } (1+\phi _2(\widehat{\xi },\widehat{\eta })) \end{array}\right) , \end{aligned}$$

(4.10)

where $\mathcal \phi _j(\widehat{\xi },\widehat{\eta }),\ j=1,2$, are ${\mathcal {C}}^{\infty }$ functions defined in a sufficiently small neighbourhood of the origin with $\mathcal \phi _j(0,0)=(0,0),\ j=1,2$.

Proof

Let $\widehat{\Phi }(\widehat{\xi },\widehat{\eta })$ be the ${\mathcal {C}}^{\infty }$ local conjugacy given by Sternberg’s Theorem. Consequently, $\widehat{\Phi }$ should send the stable and unstable manifolds of ${\mathcal {F}}$ to the corresponding ones of ${\mathcal {L}}$, which in this case means that it preserves the axes. Writing $\widehat{\Phi } = (\widehat{\Phi }_1, \widehat{\Phi }_2)$, this is translated into the conditions $\widehat{\Phi }_1(0,\widehat{\eta }) =0$ and $\widehat{\Phi }_2(\widehat{\xi },0)=0$. Then,

$$\begin{aligned} \widehat{\Phi }_1(\widehat{\xi },\widehat{\eta }) = \widehat{\Phi }_1(0,\widehat{\eta })+ \int _0^1 \partial _\xi \widehat{\Phi }_1(t \widehat{\xi },\widehat{\eta })\, \xi \, dt = \widehat{\xi }(\alpha + \widehat{\phi }_1(\widehat{\xi },\widehat{\eta }) ) \end{aligned}$$

with ${\widehat{\phi }}_1(0,0)=0$, and analogously $\widehat{\Phi }_2(\widehat{\xi },\widehat{\eta }) = \widehat{\eta }(\beta + \widehat{\phi }_2(\widehat{\xi },\widehat{\eta }) )$ with ${\phi }_2(0,0)=0$. Since $\widehat{\Phi }$ is a diffeomorphism, $\alpha \beta \ne 0$. We write $A = \left( \begin{array}{cc}\alpha & 0 \\ 0 & \beta \end{array}\right)$. We claim that $\Phi :=A^{-1}\widehat{\Phi }$ is also a conjugation from ${\mathcal {F}}$ to ${\mathcal {L}}$. Indeed, since A commutes with ${\mathcal {L}}$,

$$\begin{aligned} \Phi {\mathcal {F}}=A^{-1}\widehat{\Phi }{\mathcal {F}} = A^{-1} {\mathcal {L}} \widehat{\Phi }={\mathcal {L}} A^{-1} \widehat{\Phi }={\mathcal {L}} \Phi . \end{aligned}$$

Moreover, $\Phi$ is of the form given in (4.10). See Fig. 5. $\square$

In Lemma 4.1, we have proven the existence of the points

$$\begin{aligned} q_s\in W_{\textrm{loc}}^s\cap \widehat{W}^u \qquad \text { and} \qquad q_u \in W_{\textrm{loc}}^u\cap \widetilde{W}^s. \end{aligned}$$

Then, we can use the changes of coordinates introduced in the previous lemmas to transport those curves to a neighbourhood of the origin. Denote by $\gamma _1(t)$ and $\gamma _2(t)$ the parametrizations of $(\Phi \circ \Theta \circ {\mathcal {T}})(\widehat{W}^u)$ and $(\Phi \circ \Theta \circ {\mathcal {T}})(\widetilde{W}^s)$, respectively. We focus on the pieces of $\gamma _1(t)$ and $\gamma _2(t)$ in the first quadrant. Without loss of generality we can assume that these pieces are parametrized by $t\ge 0$.

Lemma 4.5

The curves $\gamma _1(t)$ and $\gamma _2(t)$ intersect the coordinate axes $\{\xi =0\}$ and $\{\eta =0\}$ at points $\widehat{q}_s = (0,\eta _0)$ and $\widehat{q}_u=(\xi _0, 0)$ and have a finite order contact there, respectively. Moreover, for t small enough we have that $\gamma _j(t),\ j=1,2$, admit the following parametrization

$$\begin{aligned} \begin{aligned}&\gamma _1(t)=(t^{\ell _1}(a_1+g_1(t)),\eta _0+t), \\&\gamma _2(t)=(\xi _0+t^{\ell _2}(a_2+g_2(t)),t^{\ell _3}(a_3+g_3(t))), \end{aligned} \end{aligned}$$

(4.11)

where $\ell _j\ge 1$ for $j=1,2,3$, $a_1 a_2 a_3\ne 0$, $\xi _0,\eta _0>0$, and $g_j(t)$ are $C^\infty$ functions with $g_j(0)=0$, for $j=1, 2,3$. See Fig. 6.

Proof

The lemma follows from the fact that $q_s$ and $q_u$ are points of finite order contact between the stable and the unstable manifolds of $p_1$ and the change of coordinates we have used is ${\mathcal {C}}^{\infty }$. $\square$

Fig. 6

Sketch of the situation described in Lemma 4.5 (Color figure online)

We want to show that, for k large enough, $\gamma _2(t)$ and ${\mathcal {L}}^k(\gamma _1(t))$ intersect transversally. This framework is quite close to the one in Churchill and Rod (1980, Theorem 1.1) but in their case the linear map admits the function $H(x,y)= xy$ as a first integral, which is not our case. Then, we provide a proof in our case to get the same conclusion.

Given $\lambda >1$ and $0<\mu <1$ introduced in (4.4) and (2.3) and we consider the auxiliary interpolation map

$$\begin{aligned} {\mathcal {L}}^\tau (\xi ,\eta )=\left( \begin{array}{cc} \lambda ^\tau & 0 \\ 0 & \mu ^\tau \end{array}\right) \left( \begin{array}{c} \xi \\ \eta \end{array}\right) , \qquad \tau >0. \end{aligned}$$

We also consider the first quadrant $Q= \{(\xi ,\eta ) \mid \ \xi \ge 0, \eta \ge 0\}$ and $H:Q \rightarrow Q$ defined by

$$\begin{aligned} H(\xi ,\eta )=\xi ^{\log \mu ^{-1}}\eta ^{\log \lambda }. \end{aligned}$$

(4.12)

It is continuous and real analytic in the interior of Q.

Lemma 4.6

The function H is a first integral of ${\mathcal {L}}^\tau$, $\tau >0$, in Q.

Proof

To prove the lemma, we compute

$$\begin{aligned} H\left( {\mathcal {L}}^\tau (\xi ,\eta )\right) =H\left( \lambda ^\tau \xi ,\mu ^\tau \eta \right) =\lambda ^{\tau \log \mu ^{-1}}\mu ^{\tau \log \lambda } \xi ^{\log \mu ^{-1}}\eta ^{\log \lambda } = H(\xi ,\eta ). \end{aligned}$$

$\square$

Next step is to show that there exist reparametrizations $t=\sigma _j(s)$, $j=1,2$, of the curves $\gamma _j(t),\ j=1,2$, which have a useful property.

Lemma 4.7

There exist continuous reparametrizations $\widetilde{\gamma }_j (s)=\gamma _j(\sigma _j(s))$, $s\in [0,s_0)$, of the curves $\gamma _j(t)$ given by $t=\sigma _j(s)$, $j=1,2$, that are $C^\infty$ in $(0,s_0)$ and they satisfy

$$\begin{aligned} H\left( \widetilde{\gamma }_j (s)\right) =s, \qquad s\in (0,s_0) \end{aligned}$$

for some $s_0>0$ small enough.

Proof

For $j=1$ we impose the condition $H\left( \gamma _1(t)\right) =s$ to obtain $t:=\sigma _1(s)$. Using (4.11) and (4.12) we have

$$\begin{aligned} \begin{aligned} H\left( \gamma _1(t)\right) =s&\iff t^{\ell _1\log \mu ^{-1}}(a_1+g_1(t))^{\log \mu ^{-1}}\left( \eta _0+t\right) ^{\log \lambda }=s \\&\iff G_1(t):=t(a_1+g_1(t))^{1/\ell _1}\left( \eta _0+t\right) ^{\log \lambda / (\ell _1 \log \mu ^{-1})}\\ &\quad =s^{1/\left( \ell _1\log \mu ^{-1}\right) }. \end{aligned} \end{aligned}$$

(4.13)

We have $G_1(0)=0$, $c_1:=G_1'(0)=a_1^{1/\ell _1}\eta _0^{\log \lambda / (\ell _1 \log \mu ^{-1})} \ne 0$.

Consequently, by the inverse function theorem, $G_1$ is locally invertible and we can write

$$\begin{aligned} t=\sigma _1(s):=G_1^{-1} (s^{1/(\ell _1\log \mu ^{-1})}) \end{aligned}$$

(4.14)

for $|s| < s_0$ for some $s_0 >0$ small.

For $j=2$, arguing as above, we have

$$\begin{aligned} & H\left( \gamma _2(t) \right) =s \ \iff \ G_2(t):=t(a_3+g_3(t))^{1/\ell _3}\\ & \quad \left( \xi _0+t^{\ell _2}\left( a_2+g_2(t)\right) \right) ^{\log \mu ^{-1} / (\ell _3 \log \lambda )}=s^{1/\left( \ell _3\log \lambda \right) }. \end{aligned}$$

Analogous computations imply that $c_2:= G_2'(0)=a_3^{1/\ell _3}\xi _0^{\log \mu ^{-1} / (\ell _3 \log \lambda )} \ne 0$ and

$$\begin{aligned} t=\sigma _2(s):=G_2^{-1} (s^{1/\left( \ell _3\log \lambda \right) }). \end{aligned}$$

(4.15)

$\square$

The following lemma establishes a relation between ${\mathcal {L}}^\tau \left( \widetilde{\gamma }_1(s)\right)$ and $\widetilde{\gamma }_2(s)$.

Lemma 4.8

Let $s_0>0$ be small enough. Then, there exists a $C^\infty$ function $\tau (s), \ s\in (0,s_0)$, such that

$$\begin{aligned} {\mathcal {L}}^{\tau (s)} \left( \widetilde{\gamma }_1(s)\right) =\widetilde{\gamma }_2(s), \qquad s\in (0,s_0). \end{aligned}$$

Moreover

$$\begin{aligned} \lim _{s\rightarrow 0^+} \tau (s)=\infty . \end{aligned}$$

(4.16)

In particular, there exist $k_0$ in ${\mathbb {N}}$ and a sequence of positive values $\{s_k\}_{k\ge k_0}$, such that $s_k\rightarrow 0$ and $\tau (s_k)=k$ for every $k \ge k_0$.

Proof

Let $s_0>0$ be as in Lemma 4.7. We use the following notation

$$\begin{aligned} \widetilde{\gamma }_1(s)=(\widetilde{\xi }_1(s),\widetilde{\eta }_1(s)) \qquad \text {and} \qquad \widetilde{\gamma }_2(s)=(\widetilde{\xi }_2(s),\widetilde{\eta }_2(s)). \end{aligned}$$

We define

$$\begin{aligned} \tau (s):=\frac{1}{\log \lambda }\log \frac{\widetilde{\xi }_2(s)}{\widetilde{\xi }_1(s)}, \qquad s\in (0,s_0) \end{aligned}$$

and therefore

$$\begin{aligned} \frac{\lambda ^{\tau (s)}\widetilde{\xi }_1(s)}{\widetilde{\xi }_2(s)}=1. \end{aligned}$$

(4.17)

Clearly, $\tau (s)$ is $C^\infty$ in $(0,s_0)$. Next, we will check that $\lambda ^{\tau (s)}\widetilde{\eta }_1(s)=\widetilde{\eta }_2(s)$. Using that H is a first integral and Lemma 4.7 we have that

$$\begin{aligned} H(\lambda ^{\tau }\widetilde{\xi }_1(s), \mu ^\tau \widetilde{\eta }_1(s) ) = H(\widetilde{\xi }_1(s),\widetilde{\eta }_1(s)) = s = H(\widetilde{\xi }_2(s),\widetilde{\eta }_2(s)) \end{aligned}$$

from which we deduce that

$$\begin{aligned} \left( \frac{\lambda ^{\tau }\widetilde{\xi }_1(s)}{\widetilde{\xi }_2(s)}\right) ^{\log \mu ^{-1}}=\left( \frac{\widetilde{\eta }_2(s)}{\mu ^\tau \widetilde{\eta }_1(s)}\right) ^{\log \lambda }. \end{aligned}$$

Taking $\tau =\tau (s)$, from (4.17)) we conclude

$$\begin{aligned} \mu ^{\tau (s)}\widetilde{\eta }_1(s)=\widetilde{\eta }_2(s), \end{aligned}$$

(4.18)

i.e. ${\mathcal {L}} ^\tau \left( \widetilde{\gamma }_1(s)\right) =\widetilde{\gamma }_2(s)$ as desired. Of course,

$$\begin{aligned} \lim \limits _{s\rightarrow 0^+} \tau (s)= \frac{1}{\log \lambda }\log \lim \limits _{s\rightarrow 0^+} \frac{\widetilde{\xi }_2(s)}{\widetilde{\xi }_1(s)} =\infty , \end{aligned}$$

(4.19)

since $\lim _{s\rightarrow 0^+} \widetilde{\xi }_2(s) = \xi _0>$ and $\lim _{s\rightarrow 0^+} \widetilde{\xi }_1(s) = 0$. Using and the fact that $\tau (s)$ is a $C^\infty$ function in its domain we get from Bolzano’s theorem that there exist $k_0$ such that for every $k\ge k_0$ there exists $s_k$ such that $t(s_k)=k$, and the lemma follows. $\square$

The lemma above shows that the curves ${\mathcal {L}}^k\left( \widetilde{\gamma }_1(s)\right)$ and $\widetilde{\gamma }_2(s)$ intersect at the values $s=s_k$. Next lemma shows that these intersections, for k large enough, are transversal.

Lemma 4.9

Let k be large enough. The following limits hold.

$$\begin{aligned} \lim \limits _{s\rightarrow 0^+} \frac{\lambda ^k\frac{\text {d} \widetilde{\xi }_1}{\text {d} s}(s)}{\frac{\text {d} \widetilde{\xi }_2}{\text {d} s}(s)}=\infty \qquad \text {and} \qquad \lim \limits _{s\rightarrow 0^+} \frac{\mu ^k\frac{\text {d} \widetilde{\eta }_1}{\text {d} s}(s)}{\frac{\text {d} \widetilde{\eta }_2}{\text {d} s}(s)}=0. \end{aligned}$$

(4.20)

In particular, for k large enough, the curves ${\mathcal {L}}^k\left( \widetilde{\gamma }_1(s)\right)$ and $\widetilde{\gamma }_2(s)$ intersect transversally at the values $s=s_k$.

Proof

To prove the lemma we first make some computations. From the proof of Lemma 4.7, and equations (4.14) and (4.15), we deduce the following asymptotic behaviours (as $s \rightarrow 0$)

$$\begin{aligned} \begin{array}{rlcl} \sigma _1(s) \! \! \!& = c_1^{-1} s^{1/(\ell _1\log \mu ^{-1})}+\dots ,& \quad \sigma _1^\prime (s) \! \!\!& = c_1^{-1}\frac{1}{\ell _1\log \mu ^{-1}}s^{1/(\ell _1\log \mu ^{-1})}s^{-1}+ \dots , \\ \\ \sigma _2(s) \! \!\! & = c_2^{-1} s^{1/(\ell _3\log \lambda )}+\dots ,& \quad \sigma _2^\prime (s) \! \! \! & = c_2^{-1}\frac{1}{\ell _3\log \lambda }s^{1/(\ell _3\log \lambda )}s^{-1}+\dots , \end{array}\nonumber \\ \end{aligned}$$

(4.21)

where

$$\begin{aligned} c_1 = a_1^{1/\ell _1}\eta _0^{\log \lambda / (\ell _1 \log \mu ^{-1})} \qquad \text {and} \qquad c_2 = a_3^{1/\ell _3} \xi _0 ^{\log \mu ^{-1} / (\ell _3 \log \lambda )} \end{aligned}$$

are nonzero. Hence, using (4.11), we can compute the following expressions for some of the terms of the numerator and denominator in (4.20). On the one hand, we have

$$\begin{aligned} \begin{aligned} \frac{\text {d} \widetilde{\xi }_1}{\text {d} s} (s)&=\frac{\text {d} (\xi _1\circ \sigma _1)}{\text {d} s}(s) = \sigma _1^{\ell _1-1}(s)\left[ \ell _1(a_1+g_1(\sigma _1(s)))+\sigma _1(s)g_1^{\prime }(\sigma _1(s))\right] \sigma _1^{\prime }(s) \\&=\ell _1a_1 \sigma _1^{\ell _1-1}(s) \sigma _1^{\prime }(s) +\dots = \frac{a_1}{c_1^{\ell _1}\log \mu ^{-1}}s^{1/\log \mu ^{-1}}s^{-1} +\dots , \\ \frac{\text {d} \widetilde{\xi }_2}{\text {d} s}(s)&=\frac{\text {d} (\xi _2 \circ \sigma _2)}{\text {d} s}(s) = \sigma _2^{\ell _2-1}(s)\left[ \ell _2(a_2+g_2(\sigma _2(s)))+\sigma _2(s)g_2^{\prime }(\sigma _2(s))\right] \sigma _2^{\prime }(s) \\&= \ell _2a_2 \sigma _2^{\ell _2-1}(s) \sigma _2^{\prime }(s) +\dots = \frac{\ell _2 a_2}{\ell _3 c_2^{\ell _2}\log \lambda }s^{\ell _2/(\ell _3\log \lambda )}s^{-1} + \dots . \end{aligned} \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \begin{aligned} \frac{\text {d} \widetilde{\eta }_1}{\text {d} s} (s)&=\frac{\text {d} (\eta _1 \circ \sigma _1) }{\text {d} s} (s) = \sigma ^{\prime }_1(s) =c_1^{-1} \frac{1}{\ell _1\log \mu ^{-1}} s^{1/\left( \ell _1\log \mu ^{-1}\right) }s^{-1} + \dots , \\ \frac{\text {d} \widetilde{\eta }_2}{\text {d} s} (s)&=\frac{\text {d} (\eta _2\circ \sigma _2)}{\text {d} s}(s) = \sigma _2^{\ell _3-1}(s)\left[ \ell _3(a_3+g_3(\sigma _2(s)))+\sigma _2(s)g_3^{\prime }(\sigma _2(s))\right] \sigma _2^{\prime }(s) \\&= \ell _3a_3 \sigma _2^{\ell _3-1}(s)\sigma _2^{\prime }(s) +\dots =\frac{a_3}{ c_2^{\ell _3}\log \lambda }s^{1/\log \lambda }s^{-1} + \dots . \end{aligned} \end{aligned}$$

From (4.17) to (4.18) evaluated at the values of $s= s_k$ corresponding to $\tau (s_k)=k$, we can also conclude that

$$\begin{aligned} \lambda ^k=\frac{\xi _2(\sigma _2(s_k))}{\xi _1(\sigma _1(s_k))}= \frac{\xi _0}{a_1}\sigma _1^{-\ell _1}(s_k) +\dots = \frac{\xi _0 c_1^{\ell _1}}{a_1}s^{-1/\log \mu ^{-1}}+\dots \end{aligned}$$

and

$$\begin{aligned} \mu ^k=\frac{\eta _2(\sigma _2(s_k))}{\eta _1(\sigma _1(s_k))} = \frac{a_3}{\eta _0}\sigma _2^{\ell _3}(s_k)+\dots =\frac{a_3c_2^{-\ell _3}}{\eta _0}s_k^{1/ \log \lambda } +\dots . \end{aligned}$$

All together allows us to compute the limits of the statement

$$\begin{aligned} \begin{aligned}&\lim \limits _{s\rightarrow 0^+} \lambda ^k\frac{\frac{\text {d} \widetilde{\xi }_1}{\text {d} s}(s)}{\frac{\text {d} \widetilde{\xi }_2}{\text {d} s}(s)}=\lim \limits _{s\rightarrow 0^+} \frac{\xi _0 \ell _3\log \lambda }{\ell _2a_2c_2^{-1}r_2^{\ell _2-1}\log \mu ^{-1}}s^{-\ell _2/(\ell _3\log \lambda )}=\infty \\&\lim \limits _{s\rightarrow 0^+} \mu ^k\frac{\frac{\text {d} \widetilde{\eta }_1}{\text {d} s}(s)}{\frac{\text {d} \widetilde{\eta }_2}{\text {d} s}(s)}=\lim \limits _{s\rightarrow 0^+} \frac{c_1^{-1}\log \lambda }{\eta _0 \ell _1\log \mu ^{-1}}s^{1/(\ell _1\log \mu ^{-1})}=0. \end{aligned} \end{aligned}$$

$\square$

Proof

(End of the proof of Theorem B) Let $k\ge k_0$ satisfy the conditions of the previous lemmas. We consider the sequence of points

$$\begin{aligned} \left\{ \widetilde{\gamma }_j(s_k)=\left( \widetilde{\xi }_j(s_k),\widetilde{\eta }_j(s_k)\right) \right\} _{k\ge k_0},\qquad j=1,2. \end{aligned}$$

(4.22)

From Lemma 4.9, ${\mathcal {L}}^k\left( \widetilde{\gamma }_1(s_k)\right)$ and $\widetilde{\gamma }_2(s_k)$ intersect transversally. It follows from (4.11) and the lemmas above than

$$\begin{aligned} \widetilde{\gamma }_1(s_k)\rightarrow (0,\eta _0) \qquad \text {and} \qquad \widetilde{\gamma }_2(s_k) \rightarrow (\xi _0,0). \end{aligned}$$

(4.23)

We introduce the following notation

$$\begin{aligned} & \mathbf{\widehat{z}}_k:= \left( {\mathcal {T}}^{-1}\circ \Theta ^{-1} \circ \Psi ^{-1} \right) \left( \widetilde{\gamma }_1(s_k)\right) \in \widehat{W}^u \qquad \text { and} \nonumber \\ & \mathbf{\widetilde{z}}_k:= \left( {\mathcal {T}}^{-1}\circ \Theta ^{-1} \circ \Psi ^{-1} \right) \left( \widetilde{\gamma }_2(s_k)\right) \in \widetilde{W}^s, \end{aligned}$$

(4.24)

and note that Lemma 4.8 implies that

$$\begin{aligned} \mathbf{\widetilde{z}}_k = T_d^{2k} \left( \widehat{\textbf{z}}_k\right) ,\qquad k\ge k_0. \end{aligned}$$

(4.25)

To conclude the proof of Theorem B, we argue as follows. We know that $\{\mathbf{\widehat{z}}_k\}_{k\ge k_0} \in W^u$ and $\{\mathbf{\widetilde{z}}_k\}_{k\ge k_0} \in W^s$. Also the stable and unstable manifolds are invariant sets for the map $T_d^2$. Finally, Lemma 4.9, definition (4.24) and equation (4.25) imply that $\{\mathbf{\widehat{z}}_k\}_{k\ge k_0}$ and $\{\mathbf{\widetilde{z}}_k\}_{k\ge k_0}$ correspond to transversal intersections of the stable and unstable manifolds of $p_1$ accumulating to the points $q_s$ and $q_u$, respectively. See Fig. 7. $\square$

Fig. 7

Sketch of the proof of Theorem B with the points $\mathbf{\widehat{z}}_k$ and $\mathbf{\widetilde{z}}_k$ being transversal intersections of the stable and unstable manifolds of $p_1$ (for $T_d^2$) (Color figure online)

5 Proof of Theorem C

The proof of Theorem C is based on Moser’s version of Birkhoff–Smale theorem; concretely we will apply Theorem 3.7 in Moser (2001) in our setup. The key difficulty comes from the fact that, in our case, $T_d^{-1}$ is not differentiable on the line $\{y=x\}$. Therefore, we need to make sure that the construction in Moser (2001) can be made so that we only have to deal with our map and its inverse in a domain that does not meet the line $\{y=x\}$. However, notice that both the stable and the unstable manifolds of the two-cycle $\{p_0,p_1\}$ cross the line $y=x$. See Fig. 2.

Remark 5.1

It follows from Lemma 4.1 that $W^s$ as well as $W^u$ intersect the line $\{y=x\}$ at isolated points. In other words, finite length pieces of $W^s$ and $W^u$ only contain finitely many intersections with $\{y=x\}$.

Let U be a neighbourhood of $p_1$ as in Lemma 4.4 where we can take local coordinates for which $p_1$ is located at (0, 0) and the stable and the unstable manifolds of (0, 0) are the vertical and horizontal axes, respectively. Assume also that $U\cap \{y=x\} =\emptyset$.

In the following items, we summarize notation and facts of the constructions we have made in the previous section that will be important in the proof of Theorem C. See Fig. 8.

• (a) Let q be the homoclinic point given in Theorem A and $q_s= T_d^{2\alpha }(q)\in W^s_\textrm{loc}\cap W^u$ and $q_u= T_d^{-2\beta }(q)\in W^u_\textrm{loc}\cap W^s$, $\alpha , \beta \in \mathbb {N}$, be the points given in Lemma 4.1. Let $\widehat{W}^u \subset W^u$ and $\widetilde{W}^s\subset W^s$ introduced after the statement of Lemma 4.1. Then, $q_s\in W^s_\textrm{loc}\cap \widehat{W}^u$ and $q_u\in W^u_\textrm{loc}\cap \widetilde{W}^s$. Moreover, taking $n_0= \alpha + \beta$ we have that

  $$\begin{aligned} T_d^{2n_0}\left( q_u\right) = q_s. \end{aligned}$$

• (b) Let $\widehat{\textbf{z}}_k \in \widehat{W}^u$ and $\widetilde{\textbf{z}}_k\in \widetilde{W}^s$ be the points introduced in (4.24). We have that $\widetilde{\textbf{z}}_k$ and $\widehat{\textbf{z}}_k$ are transversal homoclinic points,

  $$\begin{aligned} \lim \limits _{k\rightarrow \infty } \hat{\textbf{z}}_k =q_s \qquad \text {and} \qquad \lim \limits _{k\rightarrow \infty } \tilde{\textbf{z}}_k =q_u. \end{aligned}$$

  and

  $$\begin{aligned} T_d^{2k}\left( \widehat{\textbf{z}}_k\right) = \widetilde{\textbf{z}}_k, \qquad T_d^{2j}(\widehat{\textbf{z}}_k) \in U\quad \text { for all } \quad j=1,\ldots ,k. \end{aligned}$$

• (c) Let $\widehat{W}^s = T_d^{2n_0}(\widetilde{W}^s )$ and $\widetilde{W}^u= T_d^{-2n_0} (\widehat{W}^u)$.

• (d) We consider the points

  $$\begin{aligned} \widehat{\textbf{w}}_k:=T_d^{2n_0}(\widetilde{\textbf{z}}_k) \in \widehat{W}^s \cap W^s_\textrm{loc}\cap W^u \quad \text {and} \quad \widetilde{\textbf{w}}_k:=T_d^{-2n_0}(\widehat{\textbf{z}}_k) \in \widetilde{W}^u \cap W^u_\textrm{loc}\cap W^s. \end{aligned}$$

  As a consequence of the previous items, we have

  $$\begin{aligned} T_d^{2n_0+2k+2n_0}( \widetilde{\textbf{w}}_k)= \widehat{\textbf{w}}_k. \end{aligned}$$

• (e) For any $m_u\ge 1$ and $m_s\ge 1$, if we write,

  $$\begin{aligned} \textbf{w}_u:=T_d^{-2m_u}(\widetilde{\textbf{w}}_k)\in W^u_\textrm{loc}\qquad \text {and} \qquad \textbf{w}_s:=T_d^{2m_s}(\widehat{\textbf{w}}_k) \in W^s_\textrm{loc}, \end{aligned}$$

  we have that

  $$\begin{aligned} T_d^{2m_s+2n_0+2k+2n_0+2m_u}(\textbf{w}_u){=} \textbf{w}_s \quad \text {or} \quad T_d^{-2m_s-2n_0-2k-2n_0-2m_u}(\textbf{w}_s){=} \textbf{w}_u. \end{aligned}$$

  (5.1)

  In particular, considering $m_u$ and $m_s$ as large as necessary we know that the corresponding points $\textbf{w}_u$ and $\textbf{w}_s$ are as close as needed to the point $p_1$, and, by the $\lambda$-Lemma (Palis 1969) they are transverse homoclinic points with tangent vectors close to the tangent vectors of the local manifolds.

Proof of Theorem C

Let $U_u\subset U$ be a neighbourhood of $q_u$. Assume it is sufficiently small so that $U_s:=T_d^{2n_0}(U_u)$ is contained in U. Clearly, $U_s$ is a neighbourhood of $q_s$. From items (b) and (d), there exists $k_0>0$ such that $\widehat{\textbf{z}}_k,\widehat{\textbf{w}} _k \in U_s$ and $\widetilde{\textbf{z}}_k,\widetilde{\textbf{w}}_k \in U_u$ for all $k\ge k_0$.

Let k be large enough and let V be a small neighbourhood of $\textbf{w}_u$ (suitable size will be decided later on). Then, $T_d^{2m_u}(V)$ is a neighbourhood of $\widetilde{\textbf{w}}_k \subset U_u \subset U$ and, if we denote

$$\begin{aligned} m_0:=m_u+n_0+k_0+n_0+m_s, \end{aligned}$$

then $T_d^{2m_0}(V)$ is a neighbourhood of $\textbf{w}_s$.

Let $R\subset V$ be a pseudo-rectangle in the first quadrant attached to $\textbf{w}_u$ whose boundaries are given by pieces of $W^s$ and $W^u$ and the others are just straight lines (parallel to the tangent lines of $W^s$ and $W^u$ at $\textbf{w}_u$).

Define

$$\begin{aligned} \widetilde{R}:=T_d^{2m_0}(R). \end{aligned}$$

(5.2)

If we iterate $\widetilde{R}$ by $T_d^2$, while staying in U where the dynamics is $C^\infty$ conjugate to the one of the linearization at $p_0$, we eventually meet R.

Following Moser we introduce the transversal map $\widetilde{\phi }$. Given $\xi \in R$ we consider a number of iterates bigger that $m_0$ of $T^2_d$. By construction $T^{2m_0}_d(\xi ) \in \widetilde{R}$. Next we consider $k=k(\xi )>m_0$ to be the smallest integer such that $T_d^{2k}(T^{2m_0}_d(\xi )) \in R$ and $T_d^{2j}(T^{2m_0}_d(\xi )) \in U$, $1\le j\le k$, if it exists. We denote by ${\mathcal {D}}$ the set of $\xi \in R$ such that $k(\xi )$ exists and we define $\widetilde{\phi }: {\mathcal {D}} \subset R \rightarrow R$ by

$$\begin{aligned} \widetilde{\phi } (\xi ) = T_d^{2m_0+ 2k(\xi )}(\xi ), \qquad \xi \in {\mathcal {D}}. \end{aligned}$$

(5.3)

To apply Moser’s theorem, we should check that the restriction of $T_d$ to $\bigcup _{k=1}^{2m_0} T_d^k({\mathcal {D}})$ is a ${\mathcal {C}}^\infty$ diffeomorphism or equivalently that $\bigcup _{k=1}^{2m_0} T_d^k({\mathcal {D}}) \cap \{y=x\} =\emptyset$.

That is, we should prove that, by choosing R small enough the points travelling from R to itself would not meet the line $\{y=x\}$, where $T_d^{-1}$ is not smooth.

Since $\xi \in {\mathcal {D}} \subset R\subset V$ and V is a small neighbourhood of $\textbf{w}_u$ the iterates of $\xi \in {\mathcal {D}}$ will travel following the orbit of $\textbf{w}_u$

until they arrive to $\widetilde{R}$. Then, by item (b), from $\widetilde{R}$ to R the iterates will stay in U. Hence, the only iterates of points $q\in {\mathcal {D}} \subset R$ which might fall in the line $\{y=x\}$ are the $2n_0$ iterates needed to go from $\widetilde{\textbf{w}}_k$ to $\widehat{\textbf{z}}_k$ and the $2n_0$ ones from $\widetilde{\textbf{z}}_k$ to $\widehat{\textbf{w}}_k$. To finish the argument, we distinguish two cases.

Case 1. The finite set $\{T_d^{j}(q_u)\mid \, 0 \le j \le 2n_0\}$ does not intersect $\{y=x\}$. By continuity there exists a sufficiently small open neighbourhood $U_{u}$ of $q_u$ such that the open set

$$\begin{aligned} {\mathcal {U}}_{u}:= \bigcup _{j=0}^{2n_0} T_d^{j}(U_{u}) \end{aligned}$$

does not intersect $\{y=x\}$ either. Of course, by items (b) and (d) there are infinitely many points of the sequences $\{\widetilde{\textbf{w}}_k\}, \{\widetilde{\textbf{z}}_k\}$ belonging to ${\mathcal {U}}_{u}$.

Choose V (the neighbourhood of $\textbf{w}_u$ above) small enough and k large enough such that $T_d^{2m_u}(V) \subset U_u$ and such that $T_d^{2j}(V)$ belong to U, for $j=0,\ldots ,m_u$. Choose $R\subset V$ and define $\widetilde{R}$ as in (5.2). By construction, the map $\widetilde{\phi }$ is well defined and ${\mathcal {C}}^\infty$, it has an inverse $\widetilde{\phi }^{-1}: R\cap \widetilde{\phi }(R)$ which is also ${\mathcal {C}}^\infty$ since no iterates of $T_d$ or $T_d^{-1}$ in the definition of $\widetilde{\phi }$ (see (5.3)) intersect $\{y=x\}$.

Fig. 8

The illustration of all items a–f (Color figure online)

Case 2. The set $\{T_d^{j}(q_u)\mid \, 1\le j \le 2n_0\}$ intersects $\{y=x\}$. Let $\{T_d^{\ell _j}(q_u)\mid \, j=1,\ldots , r\}$, for $0<\ell _1<\ldots<\ell _r<2n_0$, for some $r\ge 1$, be the intersection. We recall that the stable and unstable manifolds of the point $p_1$ have discrete intersection with $\{y=x\}$. Again, items (b) and (d) imply that we can choose $k_0$ large enough such that there exist two small open neighbourhoods $U_{s}$ of $q_s$ and $U_{u}$ of $q_u$ such that for all $k\ge k_0$ we have that

$$\begin{aligned} \begin{aligned} \{T_d^{j}(U_{u}\setminus \{q_u\})\mid \, j=0,\ldots ,2n_0\} \cap \{y=x\} =\emptyset \qquad \text {and} \qquad \widetilde{\textbf{w}}_k, \widetilde{\textbf{z}}_k \in U_{u}. \end{aligned} \end{aligned}$$

We are in the same situation as in the previous case. Therefore, choosing V (the neighbourhood of $\textbf{w}_u$ above) small enough and k large enough we obtain the regularity claim for $\widetilde{\phi }$ and $\widetilde{\phi }^{-1}$.

Now, Theorem 3.7 in Moser (2001) implies that there is a Cantor set ${\mathcal {I}}$ contained in R and a homeomorphism from ${\mathcal {I}}$ to the space of sequences of N symbols ($2\le N\le \infty$) which conjugates $\widetilde{\phi }$ with the Bernoulli shift and, as a consequence, there is a dense set ${\mathcal {P}}$ of periodic orbits of $\widetilde{\phi }$, and therefore of $T_d^2$ and $T_d$ in ${\mathcal {I}}$.

Moreover, Theorem 3.8 in Moser (2001) implies that there is a dense subset ${\mathcal {H}}$ of homoclinic points to $p_1$ in ${\mathcal {I}}$. We recall from Theorem A (b) of Fontich et al. (2024) that $W^s_{p_1} \subset \partial \Omega$. Finally, since ${\mathcal {H}}\subset W^s_{p_1}$,

$$\begin{aligned} {\mathcal {P}}\subset \overline{{\mathcal {P}}}= {\mathcal {I}}= \overline{{\mathcal {H}}}\subset \overline{\partial \Omega } = {\partial \Omega }, \end{aligned}$$

that is, the boundary of $\Omega$ has infinitely many periodic orbits with arbitrary high period. $\square$