Components of Stable Isotopy Connectedness of Morse – Smale Diffeomorphisms

Abstract

In 1976 S. Newhouse, J. Palis and F. Takens introduced a stable arc joining two structurally stable systems on a manifold. Later in 1983 they proved that all points of a regular stable arc are structurally stable diffeomorphisms except for a finite number of bifurcation diffeomorphisms which have no cycles, no heteroclinic tangencies and which have a unique nonhyperbolic periodic orbit, this orbit being the orbit of a noncritical saddle-node or a flip which unfolds generically on the arc. There are examples of Morse – Smale diffeomorphisms on manifolds of any dimension which cannot be joined by a stable arc. There naturally arises the problem of finding an invariant defining the equivalence classes of Morse – Smale diffeomorphisms with respect to connectedness by a stable arc. In the present review we present the classification results for Morse – Smale diffeomorphisms with respect to stable isotopic connectedness and obstructions to existence of stable arcs including the authors’ recent results.

1 INTRODUCTION

The problem of existence of an arc which connects two structurally stable systems (Morse – Smale systems) on a manifold and which has only finitely many (or countably many) bifurcations was the 33d in the list of fifty problems in dynamical systems suggested by J. Palis and C. Pugh in [34].

In 1976 S. Newhouse, J. Palis and F. Takens in [26] introduced a stable arc connecting two structurally stable systems on a manifold. This arc preserves its qualitative properties under small disturbances. In the same year S. Newhouse and M. Peixoto [28] proved the existence of a simple arc (i. e., an arc having elementary bifurcations only) joining any two Morse – Smale flows. From the paper [8] by G. Fleitas it follows that the simple arc constructed by S. Newhouse and M. Peixoto can always be replaced by a stable one. There are examples of Morse – Smale diffeomorphisms on manifolds of any dimension which cannot be connected by a stable arc (see Section 3). Therefore, there naturally arises the problem of finding an invariant defining the equivalence class of Morse – Smale diffeomorphisms with respect to connectedness by a stable arc (stable isotopy connected component).

In the present paper we review the classification results for Morse – Smale systems with respect to stable isotopy connectedness including the recent research by the authors.

2 STABLE ARCS IN THE SPACE OF DYNAMICAL SYSTEMS

In this section we give an exact definition of the stable arc and the criterion of stability for arcs composed of diffeomorphisms with finite limit sets. Then we outline the idea of the proof of existence of a stable arc joining any two Morse – Smale flows on a given manifold.

2.1 Main Concepts of the Theory of Dynamical Systems

Let $f:M^{n}\to M^{n}$ be a diffeomorphism on a smooth closed (compact without a border) $n$-manifold ($n\geqslant 1$) $M^{n}$ with a metric $d$.

Two diffeomorphisms $f,f^{\prime}:M^{n}\to M^{n}$ are said to be topologically conjugate if there is a homeomorphism $h:M^{n}\to M^{n}$ such that $fh=hf^{\prime}$.

A point $x\in M^{n}$ is wandering for $f$ if there exists an open neighborhood $U_{x}$ of $x$ for which $f^{n}(U_{x})\cap U_{x}=\varnothing$ for every $n\in\mathbb{N}$. Otherwise $x$ is said to be nonwandering. The set of the nonwandering points of a diffeomorphism $f$ is the nonwandering set denoted by $\Omega_{f}$.

For example, every limit point of a diffeomorphism is nonwandering. Recall that a point $y\in M^{n}$ is an $\omega$-limit point for a point $x\in M^{n}$ if there is a sequence $t_{k}\to+\infty,t_{k}\in\mathbb{Z}$ such that $\lim\limits_{t_{k}\to+\infty}d(f^{t_{k}}(x),y)=0.$ The set $\omega(x)$ of all $\omega$-limit points for $x$ is its $\omega$-limit set. By changing $+\infty$ to $-\infty$ one defines the $\alpha$-limit set $\alpha(x)$ of the point $x$. The set $L_{f}=cl(\bigcup\limits_{x\in M^{n}}\omega(x)\cup\alpha(x))$ is called the limit set of the diffeomorphism $f$.

If the set $\Omega_{f}$ is finite, then each point $p\in\Omega_{f}$ is periodic. Denote by $m_{p}\in\mathbb{N}$ its period. Each periodic point $p$ has the corresponding stable and unstable manifolds defined by

$W^{s}_{p}=\{x\in M^{n}:\lim\limits_{k\to+\infty}d(f^{km_{p}}(x),p)=0\}$,

$W^{u}_{p}=\{x\in M^{n}:\lim\limits_{k\to+\infty}d(f^{-km_{p}}(x),p)=0\}$.

Both the stable and the unstable manifolds are the invariant manifolds.

On the set of periodic orbits there is a partial order relation (Smale order) defined by

$$\mathcal{O}_{p}\prec\mathcal{O}_{r}\Longleftrightarrow W^{s}_{\mathcal{O}_{p}}\cap W^{u}_{\mathcal{O}_{r}}\neq\varnothing.$$

The periodic orbits $\mathcal{O}_{1},\ldots,\mathcal{O}_{k}$ compose a cycle if $W^{s}_{\mathcal{O}_{i}}\cap W^{u}_{\mathcal{O}_{i+1}}\neq\varnothing$ for $i\in\{1,\dots,k\}$ and ${\mathcal{O}_{k+1}=\mathcal{O}_{1}}$.

A periodic point $p\in\Omega_{f}$ is said to be hyperbolic if the absolute value of each eigenvalue of the Jacobian matrix $\left(\frac{\partial f^{m_{p}}}{\partial x}\right)|_{p}$ is not equal to 1. If the absolute value of each eigenvalue is less (greater) than 1, then $p$ is a sink $($a source$)$. Sinks and sources are called nodes. If a hyperbolic point is not a node, then it is called a saddle.

It follows from the hyperbolic structure of a periodic point $p$ that its stable $W^{s}_{p}$ and unstable $W^{u}_{p}$ manifolds are the images of the spaces $\mathbb{R}^{q_{p}}$ and $\mathbb{R}^{n-q_{p}}$ by injective immersions where $q_{p}$ is the number of the eigenvalues of the Jacobian matrix whose absolute value is greater than 1. The number $\nu_{p}$ is called the orientation type of a point $p$ and it equals $+1(-1)$ if the map $f^{m_{p}}|_{W^{u}_{p}}$ preserves (reverses) the orientation of $W^{u}_{p}$. A linear connected component of the set $W^{u}_{p}\setminus p$ ($W^{s}_{p}\setminus p$) is called an unstable $($stable$)$ separatrix of the point $p$.

A closed $f$-invariant set $A\subset M^{n}$ is called an attractor of the dynamical system $f$ if $A$ has a compact neighborhood $U_{A}$ such that $f(U_{A})\subset intU_{A}$ and $A=\bigcap\limits_{k\geqslant 0}f^{k}(U_{A})$. In this case $U_{A}$ is called a trapping or an isolating neighborhood. A repeller for $f$ is the attractor for $f^{-1}$. Let $A$ be an attractor for $f$ with a trapping neighborhood $U_{A}$, $V=M^{n}\setminus int(U_{A})$ and let $R=\bigcap\limits_{k\geqslant 0}f^{-k}(V)$ be a repeller. Then the pair $A,R$ is called dual.

A diffeomorphism $f:M^{n}\to M^{n}$ is said to be a Morse – Smale diffeomorphism if

• 1) the nonwandering set $\Omega_{f}$ consists of a finite number of hyperbolic orbits;

• 2) for any two nonwandering points $p$, $q$ the manifolds $W^{s}_{p}$, $W^{u}_{q}$ transect transversally.

A Morse – Smale diffeomorphism is said to be gradient-like if from $W^{s}_{\sigma_{1}}\cap W^{u}_{\sigma_{2}}\neq\varnothing$ for any two distinct points $\sigma_{1},\sigma_{2}\in\Omega_{f}$ it follows that $dimW^{u}_{\sigma_{1}}<dimW^{u}_{\sigma_{2}}.$

A Morse – Smale flow on a manifold $M^{n}$ is defined in a similar way, and if such a flow has no periodic trajectories, it is called gradient-like.

A detailed explanation of the facts of this section can be found in [13, 16, 39].

2.2 Stability of an Arc of Diffeomorphisms

Consider a single parameter family of diffeomorphisms (an arc) $\varphi_{t}:M^{n}\to M^{n},t\in[0,1]$. The arc $\varphi_{t}$ is called smooth if the map $F:M^{n}\times[0,1]\to M^{n}$ defined by $F(x,t)=\varphi_{t}(x)$ is a diffeotopy, i. e., a smooth map which is a diffeomorphism for each fixed $t$. This map is called the isotopy in the topological category.

The smooth arc $\varphi_{t}$ is called the smooth composition of the smooth arcs $\phi_{t}$ and $\psi_{t}$ if $\phi_{1}=\psi_{0}$ and $\varphi_{t}=\begin{cases}\phi_{\tau(2t)},0\leqslant t\leqslant\frac{1}{2},\\ \psi_{\tau(2t-1)},\frac{1}{2}\leqslant t\leqslant 1,\end{cases}$ where $\tau:[0,1]\to[0,1]$ is a smooth monotonic map such that $\tau(t)=0$ for $0\leqslant t\leqslant\frac{1}{3}$ and $\tau(t)=1$ for $\frac{2}{3}\leqslant t\leqslant 1$. We write

$$\varphi_{t}=\phi_{t}*\psi_{t}.$$

Following [27] a smooth arc $\varphi_{t}$ is called stable if it is an inner point of the equivalence class with respect to the following relation: two arcs $\varphi_{t}$, $\varphi^{\prime}_{t}$ are said to be conjugate if there are homeomorphisms $h:[0,1]\to[0,1],H_{t}:M\to M$ such that $H_{t}\varphi_{t}=\varphi^{\prime}_{{h(t)}}H_{t},t\in[0,1]$ and $H_{t}$ continuously depends on $t$.

Denote by $\mathcal{Q}$ the set of smooth arcs $\varphi_{t},t\in[0,1]$ such that the starting point and the ending point of arcs of $\mathcal{Q}$ is a Morse – Smale diffeomorphism and every diffeomorphism $\varphi_{t}$ has a finite limit set.

It is shown in [27] that an arc $\varphi_{t}\in\mathcal{Q}$ for $t\in[0,1]$ is stable if and only if its every point is a stable diffeomorphism except for a finite number of bifurcation points $\varphi_{b_{i}},i=1,\dots,q$ for which

• 1) the limit set of the diffeomorphism $\varphi_{b_{i}}$ contains a unique nonhyperbolic periodic orbit which is a saddle-node or a flip;

• 2) the diffeomorphism $\varphi_{b_{i}}$ has no cycles;

• 3) the invariant manifolds of all periodic points of $\varphi_{b_{i}}$ transect transversally;

• 4) $\varphi_{b_{i}}$ has a unique nonhyperbolic periodic orbit which is the orbit of a noncritical saddle-node or of a flip which unfolds generically (Fig. 1).

Fig. 1

A stable arc.

Recall the definition of a fixed saddle-node or a flip generic unfolding when the nonhyperbolic point is of period $k>1$. For the arc $\varphi^{k}_{t}$ the definition is similar.

A saddle-node $p$ unfolds generically on the arc $\varphi_{t}\in\mathcal{Q}$ where $t\in[0,1]$ (Fig. 2) if in some neighborhood of the point $(p,b_{i})\in M^{n}\times[0,1]$ the arc $\varphi_{t}$ (or the arc $\varphi_{1-t}$) is conjugate to the arc

$$\displaystyle\tilde{\varphi}_{\tilde{t}}(x_{1},x_{2},\ldots,x_{1+n_{u}},x_{2+n_{u}},\ldots,x_{n})$$

$$\displaystyle=\left(x_{1}+\frac{x_{1}^{2}}{2}+\tilde{t},\pm 2x_{2},\ldots,\pm 2x_{1+n_{u}},\frac{\pm x_{2+n_{u}}}{2},\ldots,\frac{\pm x_{n}}{2}\right),$$

where $(x_{1},\ldots,x_{n})\in\mathbb{R}^{n},|x_{i}|<1/2,|\tilde{t}|<1/10$.

In the local coordinates $(x_{1},\dots,x_{n},\tilde{t})$ the bifurcation occurs at the time $\tilde{t}=0$, the coordinate origin $O\in\mathbb{R}^{n}$ being the saddle-node (Fig. 3). Here the axis $Ox_{1}$ is the central manifold, the set $\{(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}:x_{1}\geqslant 0,x_{2+n_{u}}=\ldots=x_{n}=0\}$ is the unstable manifold of $O$, the set $\{(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}:x_{1}\leqslant 0,x_{2}=\ldots=x_{1+n_{u}}=0\}$ is the stable manifold of $O$.

Fig. 2

The saddle-node bifurcation.

Fig. 3

The graphs of the map $x_{1}+\frac{x_{1}^{2}}{2}+\tilde{t}$ for $\tilde{t}=-0,1;\tilde{t}=0$ and $\tilde{t}=0,1$.

If $p$ is a saddle-node of the diffeomorphism $\varphi_{b_{i}}$, then there is a unique $\varphi_{b_{i}}$-invariant foliation $F^{ss}_{p}$ with smooth leaves such that $\partial W^{s}_{p}$ is a leaf of $F^{ss}_{p}$ [18]. $F^{ss}_{p}$ is called the strongly stable foliation (Fig. 4). The analogous strongly unstable foliation is denoted by $F^{uu}_{p}$. A point $p$ is called $s$-critical if there exists a hyperbolic periodic point $q$ such that $W^{u}_{q}$ intersects some leaf of $F^{ss}_{p}$ nontransversally; the $u$-criticality is defined analogously. The point $p$ is called

• semicritical if it is either $s$- or $u$-critical;

• bicritical if it is both $s$- and $u$-critical;

• noncritical if it is neither $s$- nor $u$-critical.

For the first time instability of an arc in the neighborhood of a critical saddle-node was found in 1974 by V. S. Afraimovich and L. P. Shilnikov [37, 36]. The existence of invariant foliations $F^{ss}_{p},F^{uu}_{p}$ was previously proved by V. I. Lukyanov and L. P. Shilnikov in [20].

Fig. 4

The strongly stable and unstable foliations.

Remark 1.

For surface diffeomorphisms the stable and the unstable manifolds of a saddle-node $p$ are of dimension one and two. If $p$ is noncritical, then its invariant 1-manifold does not intersect separatrices of saddles. The 1-dimensional foliation of the 2-manifold of the saddle-node must transversally intersect the separatrices of the saddles $($Fig. 5$)$.

Fig. 5

$p_{1}$ is a $s$-critical saddle-node, $p_{2}$ is an $u$-critical saddle-node, $p_{3}$ is a noncritical saddle-node.

A flip $p$ unfolds generically on the arc $\varphi_{t}\in\mathcal{Q}$ where $t\in[0,1]$ (Fig. 6) if in some neighborhood of the point $(p,b_{i})$ the arc $\varphi_{t}$ (or the arc $\varphi_{1-t}$) is conjugate to the arc

$$\displaystyle\tilde{\varphi}_{\tilde{t}}(x_{1},x_{2},\dots,x_{1+n_{u}},x_{2+n_{u}},\ldots,x_{n})$$

$$\displaystyle=\left(-x_{1}(1\pm\tilde{t})+x_{1}^{3},\pm 2x_{2},\ldots,\pm 2x_{1+n_{u}},\frac{\pm x_{2+n_{u}}}{2},\ldots,\frac{\pm x_{n}}{2}\right),$$

where $(x_{1},\ldots,x_{n})\in\mathbb{R}^{n},|x_{i}|<1/2,|\tilde{t}|<1/10$ (Fig. 7).

Fig. 6

The period-doubling bifurcation (flip).

Fig. 7

The graphs of the map $-x_{1}(1\pm\tilde{t})+x_{1}^{3}$ and its square for $\tilde{t}=-0,1;\tilde{t}=0$ and $\tilde{t}=0,1$.

We say two diffeomorphisms $f_{0},f_{1}$ are of the same class of stable isotopy connectedness if in the space of diffeomorphisms they can be joined by an arc satisfying the previously described properties 1–4.

2.3 A Simple Arc Joining Two Morse – Smale Flows

The problem of existence of a “good” arc joining two Morse – Smale flows on a given manifold was solved by S. Newhouse and M. Peixoto in [28].

Theorem 1([28], Theorem B).

Any two Morse – Smale flows on a given manifold can be joined by a simple arc.

The simplicity means that the entire arc consists of Morse – Smale systems except for a finite set of points at which the vector field deviates from the Morse – Smale field in the least way in some sense. More exactly, at these points the vector field either has a unique nonhyperbolic saddle-node or it has a unique curve which is the nontransversal intersection of the invariant manifolds of the saddles (heteroclinic tangency).

The idea of the proof is to construct an arc joining the initial flow with the gradient flow of some Morse function.

Let $M^{n}$ be a smooth $n$-manifold and $\Phi:M^{n}\to\mathbb{R}$ be a $C^{r}$-smooth ($r\geqslant 2$) function. The point $p\in M^{n}$ is called the critical point of $\Phi$ if $grad\Phi(p)=0$, i. e., $\frac{\partial{\Phi}}{\partial{x}_{1}}(p)=\ldots=\frac{\partial{\Phi}}{\partial{x}_{n}}(p)=0$ in the local coordinates $x_{1},\dots,x_{n}$ of $p$. A critical point $p$ is nondegenerate if the matrix of the second-order partial derivatives (Hessian matrix) $\left(\frac{\partial^{2}{\Phi}}{\partial{x}_{i}\partial{x}_{j}}\right)|_{p}$ is nonsingular. Otherwise $p$ is said to be degenerate. The function $\Phi$ is called the Morse function if it has no degenerate critical points.

A continuous function $\Phi:M^{n}\to\mathbb{R}$ is called a Lyapunov function for a Morse – Smale system (a diffeomorphism or a flow) on the manifold $M^{n}$ if it strictly decreases along the wandering trajectories and is constant on the periodic orbits.

A smooth Lyapunov function $\Phi$ is called an energy function for a Morse – Smale system if the set of the critical points of $\Phi$ coincides with the nonwandering set of the system.

When constructing the desired arc the first step would be to destroy the closed orbits by sequentially creating a saddle-node on each closed trajectory (Fig. 8). Then when passing through the saddle-node there appear two hyperbolic points of neighboring indices, while the end point of the arc is the gradient-like flow.

Fig. 8

The destruction of periodic orbits.

By Franks’ lemma in [9] a gradient-like flow can be connected by an arc without bifurcations to a gradient-like flow which is locally gradient in the neighborhood of its fixed points. S. Smale in [38] proved that any such flow allows a Morse energy function. Then using the level curves of this function one constructs an arc joining the gradient-like flow to the gradient flow of its Morse function (see Section 4.2 where this idea is used for Morse – Smale diffeomorphisms). The last step is to join the two Morse functions by an arc which typically generates the simple arc of gradient flows.

Unfortunately, the results of S. Newhouse and M. Peixoto cannot be directly used to construct a stable arc between two Morse – Smale diffeomorphisms. The first reason is that typically Morse – Smale diffeomorphisms cannot be included in Morse – Smale flows (see, for example, [10, 11] and [12]). The other reason is that the discretization (i. e., replacement of the flows by their 1-time shifts) of an arc with a heteroclinic tangency is not a stable arc. The last problem is solved by the following theorem due to G. Fleitas [8].

Theorem 2 ([8 ], Theorem).

If two gradient-like flows on the manifold $M^{n}$ are joined by an arc with the unique bifurcation point of heteroclinic tangency, then they can be joined by an arc with two saddle-node bifurcations $($ Fig. 9 $)$ . The discretization of this arc is a stable arc that joins $1$ -time shifts of the initial gradient-like flows.

Fig. 9

Replacing heteroclinic tangencies by saddle-nodes.

3 THE OBSTRUCTIONS TO THE EXISTENCE OF A STABLE ARC BETWEEN TWO ISOTOPIC MORSE – SMALE DIFFEOMORPHISMS

In this section we review the well-known obstructions to the existence of a stable arc between two isotopic Morse – Smale diffeomorphisms.

3.1 The Discrepancy Between Rotation Numbers of Two Rough Transformations of a Circle

The rotation number was introduced by H. Poincaré for homeomorphisms of a circle while studying flows on the torus without fixed points.

Consider the circle $S^{1}=\mathbb{R}/\mathbb{Z}$, an orientation-preserving homeomorphism $f:S^{1}\to S^{1}$ and its lift $\bar{f}:\mathbb{R}\to\mathbb{R}$ with respect to the covering map

$$\pi(x)=\{x\},$$

where $\{x\}$ is the fractional part of $x\in\mathbb{R}$. Then for every $x\in\mathbb{R}$ there exists the limit

$$\lim_{n\to\infty}\frac{\bar{f}^{n}(x)-x}{n}$$

whose fractional part $r(f)$ is independent of the choice of the lift of $f$ as well as of the point $x$. It is called the Poincaré rotation number.

Morse – Smale diffeomorphisms of the circle were thoroughly studied by A. G. Mayer [22]. He showed that, if a circle transformation is rough, then it is a Morse – Smale diffeomorphism (a diffeomorphism with a finite number of hyperbolic periodic points). Any such diffeomorphism is a composition of the 1-time shift of a gradient flow of some Morse function and either an orientation-changing involution for an orientation-changing diffeomorphism or a rational rotation for an orientation-preserving diffeomorphism.

Since the rotation number continuously depends on the homeomorphism in the $C^{0}$ topology (see, for example, [19]), any arc joining two orientation-preserving Morse – Smale diffeomorphisms with distinct rotation numbers contains a continuum of bifurcations and, therefore, it is not stable.

Theorem 3 ([30], Theorem 1).

All rough orientation-reversing diffeomorphisms of the circle lie in the same component of the stable isotopy connectedness, whereas the stable isotopy class of the rough orientation-preserving transformation of the circle is completely determined by the Poincaré rotation number.

The idea of the proof is to construct a bifurcation-free arc joining an arbitrary diffeomorphism of a given topological conjugacy class to the corresponding model diffeomorphism $\Phi_{n,m,k}$ or $\Psi_{q,\nu}$ (Fig. 10) constructed as follows.

Fig. 10

(a) Orientation-preserving diffeomorphisms of the circle; (b) orientation-reversing diffeomorphisms of the circle.

Let $(n,m,k)$ be a triple of integers such that either $n\in\mathbb{N}$ and $k=0$ for $m=1$ or $k\in\{1,\dots,m-1\}$ when $m>1$ and $(m,k)$ are mutually prime. Consider the map $\bar{\Phi}_{n,m,k}:\mathbb{R}\to\mathbb{R}$ defined by (Fig. 11):

$$\bar{\Phi}_{n,m,k}(x)=x+\frac{1}{4nm\pi}\sin(2nm\pi x)+\frac{k}{m}.$$

Let $\Phi_{n,m,k}=\pi\bar{\Phi}_{n,m,k}\pi^{-1}:S^{1}\to S^{1}$. Then $\Phi_{n,m,k}$ is an orientation-preserving diffeomorphism with the rotation number $\frac{k}{m}$ and $2n$ periodic orbits (half of which are attracting and the other half are repelling) of period $m$.

Fig. 11

The map $\bar{\Phi}_{n,m,k}(x)-\frac{k}{m}$.

Let $(q,\nu)$ be a pair of numbers $q\in\mathbb{N}$, $\nu\in\{-1,0,+1\}$ and let $\nu=0$ if and only if $q$ is odd. Consider the maps $\bar{\Psi}_{q,\nu}:\mathbb{R}\to\mathbb{R}$ (Fig. 12) defined by

$$\displaystyle\Psi_{q,\nu}(x)=-x-\frac{1}{4\pi q}\sin(2\pi qx),\nu=0;-1;$$

$$\displaystyle\Psi_{q,+1}(x)=\Psi^{-1}_{q,-1}\left(x-\frac{1}{2q}\right)+\frac{1}{2q},q=2\kappa,\kappa\in\mathbb{N}.$$

Let $\Psi_{q,\nu}=\pi\bar{\Psi}_{q,\nu}\pi^{-1}:S^{1}\to S^{1}$. Then $\Psi_{q,\nu}$ is an orientation-reversing diffeomorphism with $2q$ periodic orbits, two of which $(1,0),(-1,0)$ are fixed points and the other $2(q-1)$ are periodic orbits with period 2. Thus, $\Psi_{2\kappa-1,0}$ corresponds to the case when the fixed points have opposite stability, and $\Psi_{2\kappa,\pm 1}$ corresponds to the case where the fixed points are both unstable (-1) or both stable (+1), respectively.

Fig. 12

The maps $\Psi_{q,\nu}$: 1) $\bar{\Psi}_{2,1}$, 2) $\bar{\Psi}_{2,-1}$, 3) $\bar{\Psi}_{3,0}$.

Thus, the problem is reduced to the problem of finding the classes of the stable isotopy connectedness for the model diffeomorphisms.

For the orientation-preserving diffeomorphism $\Phi_{n,m,k},n>1$ the number of periodic orbits can be reduced by one pair if one constructs the arc passing through the noncritical saddle-node which unfolds generically. Therefore, the diffeomorphism $\Phi_{n,m,k}$ can be joined by a stable arc to the diffeomorphism $\Phi_{1,m,k}$ with the same rotation number. Since the rotation number is a topological invariant of circle homeomorphism and since it continuously depends on the parameter of the arc, any arc joining two orientation-preserving circle diffeomorphisms $f,f^{\prime}$ with distinct rotation numbers is not stable because it contains a continuum of bifurcations, and that contradicts the definition of the stable arc.

The orientation-reversing diffeomorphism $\Psi_{2\kappa-1,0},\kappa>1$ (as in the case of orientation-preserving diffeomorphism) can be joined to the “source-sink” diffeomorphism $\Psi_{1,0}$ by an arc with $(2\kappa-3)$ noncritical saddle-nodes unfolding generically (this applies to period-2 orbits). For the diffeomorphism $\Psi_{q,\pm 1}$ the number $q$ is odd and $q>2$. The described technique allows one to join the diffeomorphism $\Psi_{2\kappa,\pm 1},\kappa>1$ to the diffeomorphism $\Psi_{2,\pm 1}$ which, in turn, is joined to the “source-sink” diffeomorphism $\Psi_{1,0}$ by an arc with period-doubling bifurcations.

3.2 The Nontrivial Relatedness of Periodic Points

In [21] S. Matsumoto showed that the 2-torus $\mathbb{T}^{2}$ allows Morse – Smale diffeomorphisms that are isotopy equivalent to the identity, but cannot be joined by a stable arc.

Two periodic points $p,q$ of a diffeomorphism $f:M^{n}\to M^{n}$ are said to be trivially related if there is an arc $c\subset M^{n}$ for which $\partial c=\{q\}-\{p\}$ and for some integer $N$ the closed curve $f^{N}(c)-c$ is null-homotopic, $f^{N}(p)=p$ and $f^{N}(q)=q$. Otherwise the points $p,q$ are nontrivially related. Notice that this property is independent of the choice of the curve $c$ if $f$ is isotopy equivalent to the identity. If all periodic points of $f$ are trivially related, then $f$ is trivial. Otherwise $f$ is nontrivial.

S. Matsumoto constructed two Morse – Smale diffeomorphisms $f_{0},f_{1}:\mathbb{T}^{2}\to\mathbb{T}^{2}$, both being isotopy equivalent to the identity and such that $f_{0}$ is the 1-time shift of the gradient flow of a typical Morse function with 4 critical points, while $f_{1}$ is the composition of $f_{0}$ with the two oppositely directed Dehn twists (Fig. 13). One can easily see that $f_{0}$ is trivial, while $f_{1}$ is nontrivial.

Fig. 13

S. Matsumoto’s example.

Theorem 4 ([21], Theorem 1.3).

The diffeomorphisms $f_{0}$, $f_{1}$ of the $2$-torus $\mathbb{T}^{2}$ cannot be joined by a stable arc ¹ .

To explain this result, suppose the contrary, i. e., that there is a stable arc that joins trivial and nontrivial diffeomorphisms. Then there is a stable arc $\phi_{t}$ with the unique bifurcation $g=\phi_{\frac{1}{2}}$ and such that $\phi_{0}$ is trivial, while $\phi_{1}$ is nontrivial. Since the period-doubling bifurcation does not produce nontrivial relations between the periodic points, the diffeomorphism $g$ must have a saddle-node periodic point; denote it by $p$. Denote by $\mathcal{O}_{p}$ its orbit and denote by $\mathcal{O}_{q},\mathcal{O}_{r}$ the nearest with respect to the Smale order neighboring orbits (possibly not unique). Since $g$ has no cycles, these orbits are in the partial Smale order

$$\mathcal{O}_{q}\preceq\mathcal{O}_{p}\preceq\mathcal{O}_{r},$$

i.e., $W^{s}_{\mathcal{O}_{q}}\cap W^{u}_{\mathcal{O}_{p}}\neq\varnothing$, $W^{s}_{\mathcal{O}_{p}}\cap W^{u}_{\mathcal{O}_{r}}\neq\varnothing$ and there is no such periodic point $x$ for which either $W^{s}_{\mathcal{O}_{q}}\cap W^{u}_{\mathcal{O}_{x}}\neq\varnothing$, $W^{s}_{\mathcal{O}_{x}}\cap W^{u}_{\mathcal{O}_{p}}\neq\varnothing$ or $W^{s}_{\mathcal{O}_{p}}\cap W^{u}_{\mathcal{O}_{x}}\neq\varnothing$, $W^{s}_{\mathcal{O}_{x}}\cap W^{u}_{\mathcal{O}_{r}}\neq\varnothing$.

Since the saddle-node $p$ is noncritical, either $q$ or $r$ is a node. Suppose that an unstable separatrix $l$ of a saddle-node point $p$ is attracted by the sink $q$. Then consider a curve $c=cl(l)$ which is invariant under some power $N$ of $f$ such that $f^{N}(p)=p$ and $f^{N}(q)=q$ and, hence, $f^{N}(c)-c$ is null-homotopic. As the trivial relatedness is transitive, we find that $g$ is trivial. On the other hand, the saddle-node bifurcation creates new periodic points that are trivially related to the old ones and, hence, $\phi_{1}$ is trivial and we come to a contradiction.

Obviously, similar effects are intrinsic to all surfaces with the nontrivial fundamental group. Moreover, they are naturally generalized to manifolds of greater dimensions. In [7] the trivial $f_{0}$ and the nontrivial $f_{1}$ Morse – Smale diffeomorphisms are constructed on the manifold $S^{n-1}\times S^{1},n\geqslant 3$, both being isotopy equivalent to the identity (Fig. 14). As in Matsumoto’s example, the diffeomorphism $f_{0}$ is the Cartesian product of the “source-sink” diffeomorphisms on the sphere ${S}^{n-1}$ and on the circle $S^{1}$. A diffeomorphism $f_{1}$ diffeotopic to the identity is constructed from $f_{0}$ by composition with the following multidimensional Dehn twist $\Gamma$. Namely, consider an $(n-1)$-annulus $K=S^{n-2}\times S^{1}$ in $S^{n-1}\times S^{1}$. Then the set $W\subset S^{n-1}\times S^{1}$ bounded by $K$ and $f_{0}(K)$ is a fundamental domain of $f_{0}$ on $S^{n-1}\times S^{1}\setminus(cl(W^{u}_{\sigma_{1}}\cup W^{s}_{\sigma_{2}}))$ and $W\cong S^{n-2}\times[0,1]\times S^{1}$. Let $\Gamma:S^{n-1}\times S^{1}\to S^{n-1}\times S^{1}$ be the identity outside $W$, and on $W$ let it be defined by $\Gamma(s,t,e^{i2\pi\varphi})=(s,t,e^{i2\pi(\varphi+t)}).$

Fig. 14

An example of diffeomorphisms on ${S}^{n-1}\times S^{1},n\geqslant 3$ which cannot be joined by a stable arc.

Theorem 5 ([7], Theorem 1).

The diffeomorphisms $f_{0}$ , $f_{1}$ on the manifold ${S}^{n-1}\times S^{1},n\geqslant 3$ cannot be joined by a stable arc.

3.3 The Inconsistency of the Periodic Decompositions

D. Pixton in [35] proved that an energy Morse function $\Phi_{f}:M^{2}\to\mathbb{R}$ exists for any Morse – Smale diffeomorphism $f$ on a surface $M^{2}$. P. Blanchard [1] constructed the special decomposition of $M^{2}$ by the level curves of $\Phi_{f}$ using the notion of the oddness of a periodic orbit. He proved that the consistency of these decompositions for distinct diffeomorphisms is the necessary condition of existence of a stable arc between them.

Let $M^{2}$ be an orientable surface and $f:M^{2}\to M^{2}$ be an orientation-preserving Morse – Smale diffeomorphism with the nonwandering set $\Omega_{f}$. The oddness of the periodic orbit $\mathcal{O}_{p}\subset\Omega_{f}$ of period $m$ is the odd integer $l$ for which $m=2^{k}l$ for some nonnegative $k$. A decomposition

$$M^{2}=M_{1}\cup\dots\cup M_{n}$$

of $M^{2}$ is said to be a periodic decomposition if it satisfies:

• each element $M_{i}$ is a surface whose boundary is either empty or consists of the regular level curves of $\Phi_{f}$; distinct submanifolds $M_{i}$ and $M_{j}$ may intersect only along their common boundary component;

• $\Omega_{f}\cap M_{i}\neq\varnothing$ for each $i$; all the periodic orbits $\mathcal{O}_{p}\subset M_{i}$ are of the same oddness $l_{i}$, called the oddness of $M_{i}$; if $M_{i}\cap M_{j}\neq\varnothing$ for $i\neq j$, then their oddnesses are different.

An element $M_{i}$ of the periodic decomposition is called inessential if it is homeomorphic to a union of annuli and for each annulus the vector field $\Phi_{f}$ points out along one boundary component and it points in along the other boundary component.

The elementary cancellation of a periodic decomposition is the elimination of an inessential element $M_{i}$ from the decomposition by uniting it with the element $M_{j}$ which has the common boundary with the annulus and such that $grad\Phi_{f}$ points out of the annulus along this boundary. Then the new element is assigned the oddness $l_{j}$ and is united with the elements of the same oddness with which it has the common border. Notice that the result of an elementary cancellation is not a periodic decomposition.

Denote by $\mathcal{D}_{f}$ the decomposition of the surface $M^{2}$ obtained from a periodic decomposition by all elementary cancellations. Two decompositions $\mathcal{D}_{f},\mathcal{D}_{f^{\prime}}$ of the surface $M^{2}$ are equivalent if there is a homeomorphism of $M^{2}$ which sends the elements of one decomposition to the elements of the other and preserves the oddness. Notice that decompositions corresponding to different energy functions for the same diffeomorphism $f$ are equivalent.

Theorem 6 ([1], Theorem 2.13).

If two Morse – Smale diffeomorphisms $f,f^{\prime}:M^{2}\to M^{2}$ can be joined by a stable arc, then their respective decompositions $\mathcal{D}_{f},\mathcal{D}_{f^{\prime}}$ are equivalent.

The idea of the proof is based on the fact that for the saddle-node bifurcation and the flip bifurcation the periodic decomposition of the diffeomorphism after the bifurcation is equivalent to the elementary cancellation of the periodic decomposition of the diffeomorphism before the bifurcation.

One can easily construct infinitely many diffeomorphisms with pairwise nonequivalent decompositions. Therefore, in every isotopy class containing an orientation-preserving Morse – Smale diffeomorphism there are infinitely many classes of stable isotopic connectedness.

For illustration consider the following diffeomorphisms of the sphere $S^{2}$:

1) $h$ is the “source-sink” diffeomorphism (Fig. 15.1);

2) $g$ is the Morse – Smale diffeomorphism whose nonwandering set $\Omega_{g}$ consists of the fixed sink $\omega_{1}$, the fixed source $\alpha$, the sink orbit of period $m$: $\{\omega_{2},f(\omega_{2}),\ldots,f^{m-1}(\omega_{2})\}$ and the saddle orbit of period $m$: $\{\sigma,f(\sigma),\ldots,f^{m-1}(\sigma)\}$ (Fig. 15.2 for $m=3$);

3) $f$ is the diffeomorphism with two fixed sources at the poles, which on the equator of the sphere coincides with the rough transformation of the circle with two periodic orbits of period $m$ (Fig. 15.3 for $m=3$).

Fig. 15

Examples of diffeomorphisms on the sphere.

Consider the respective periodic decompositions for these diffeomorphisms (Fig. 15):

1) $M_{1}=S^{2}$;

2), 3) $S^{2}=D_{1}\cup C\cup D_{2}$ where $D_{1}$, $D_{2}$ are the 2-disks and $C$ is the 2-annulus.

One can check that, though the periodic decompositions of the diffeomorphisms $h$ and $g$ are different, the annulus $C$ in the decomposition of $g$ is inessential. The vector field $grad\Phi_{g}$ enters from one side of $C$ and leaves from the other. After the elementary cancellation the annulus is united with the disk $D_{1}$ and the two remaining disks are united along the border producing the sphere $S^{2}$. Thus, the decompositions $\mathcal{D}_{h}$ and $\mathcal{D}_{g}$ are equivalent, which is not surprising because $h$ and $g$ can be joined by a stable arc with one saddle-node bifurcation.

The annulus $C$ in the decomposition of $f$ is not inessential, its periodic decomposition cannot be reduced. Therefore, the decompositions $\mathcal{D}_{f}$ and $\mathcal{D}_{g}$ are not equivalent and the diffeomorphisms $f$ and $g$ are in different classes of stable connectedness.

Notice that diffeomorphisms like $f$ with the distinct oddnesses of periodic orbits cannot be joined by a stable arc, and this produces infinitely many classes of stable isotopy connectedness. Nevertheless, diffeomorphisms with the same oddness but with distinct periods of periodic orbits (for example, $m=3$ and $m=6$) have equivalent periodic decompositions. We will see in Section 4.1 that such diffeomorphisms cannot be joined by a stable arc, therefore, the inverse of Theorem 6 does not hold true.

3.4 The Nonhomeomorphic Characteristic Spaces

V. Grines and O. Pochinka found ([14]) a new obstruction to the existence of a stable arc. To explain it let us represent the dynamics of any Morse – Smale diffeomorphism in the following way (see, for example, [2, 5, 13] and Chapter 2.2 of [16]).

Denote by $\Omega_{f}^{q},q=0,1,2,3$ the set of periodic points $p$ such that $\dim W^{u}_{p}=q$. Then $A_{f}=W^{u}_{\Omega_{f}^{0}\cup\Omega_{f}^{1}}$ is the connected attractor and $R_{f}=W^{s}_{\Omega_{f}^{3}\cup\Omega_{f}^{2}}$ is the connected repeller, their topology dimensions being less or equal to 1. The sets $A_{f}$ and $R_{f}$ do not intersect; every point of the set $V_{f}=M^{3}\setminus\left(A_{f}\cup R_{f}\right)$ is wandering and moves under $f$ from $R_{f}$ to $A_{f}$. The orbit space $\hat{V}_{f}=V_{f}/f$ is called the characteristic space. It is proved (see, for example, Theorem 1.2 of [17]) that a characteristic space is a simple manifold ² . Denote by $p_{{}_{f}}:V_{f}\to\hat{V}_{f}$ the natural projection.

A heteroclinic intersection of a Morse – Smale diffeomorphism is an intersection of the invariant manifolds of distinct saddle points of this diffeomorphism.

Theorem 7 ([14], Lemma 1).

Let diffeomorphisms $f,f^{\prime}:M^{3}\to M^{3}$ have no heteroclinic points and let them be joined by a stable arc. Then their characteristic spaces $\hat{V}_{f},\hat{V}_{f^{\prime}}$ are homeomorphic.

The idea of the proof is based on the fact that the dynamics in the trapping neighborhood of either attractor or repeller $R_{f}$ does not change when passing though a bifurcation point of the stable arc if this bifurcation is not the appearance or the disappearance of a heteroclinic curve. Therefore, the topology of the characteristic space remains unchanged.

To illustrate Theorem 7, we first consider the simplest Morse – Smale diffeomorphism “source-sink”. Its nonwandering set consists of two points: the source and the sink. The ambient manifold is homeomorphic to the sphere.

Theorem 8 ([4], Theorem 1).

Any two source-sink diffeomorphisms on $S^{3}$ can be joined by a bifurcation-free arc.

Notice that for a source-sink diffeomorphism on the 3-sphere the attractor $A_{f}$ consists of the unique sink $\omega$, while the repeller $R_{f}$ consists of the unique source $\alpha$. Since the diffeomorphism $f|_{W^{s}_{\omega}}$ is topologically conjugate in $\mathbb{R}^{3}$ to a homothety, the manifold $\hat{V}_{f}=\hat{V}_{\omega}=(W^{s}_{\omega}\setminus\omega)/f$ is homeomorphic to the manifold $S^{2}\times S^{1}$. So, the attractor and the repeller of a source-sink diffeomorphism are separated by a 2-sphere in the following sense.

We say that the attractor $A_{f}$ and a repeller $R_{f}$ of a Morse – Smale diffeomorphism $f:M^{3}\to M^{3}$ are separated by a 2-sphere if there is a smooth 2-sphere $\Sigma_{f}\subset V_{f}$ such that $A_{f}$ and $R_{f}$ are in the different connected components of the set $M^{3}\setminus\Sigma_{f}$ (Fig. 16).

Fig. 16

A Morse – Smale diffeomorphism $f:M^{3}\to M^{3}$ whose attractor $A_{f}$ and repeller $R_{f}$ are separated by a 2-sphere.

Theorem 9 ([14], Theorem 1).

A Morse – Smale diffeomorphism $f:M^{3}\to M^{3}$ without heteroclinic intersections can be joined to the source-sink diffeomorphism by a stable arc if and only if its attractor $A_{f}$ and repeller $R_{f}$ are separated by a $2$ -sphere.

The key technical point for the proof of Theorem 9 is the following fact. Any diffeomorphism of the considered class (distinct from source-sink diffeomorphism) has a node point whose basin contains a unique saddle separatrix. This separatrix is one-dimensional and has the same period as the node point. Moreover, it is tamely embedded to the basin in the sense of the following definition.

Consider a Morse – Smale diffeomorphism $f:M^{3}\to M^{3}$ with a fixed saddle point $\sigma$ whose 1-dimensional unstable saddle separatrix $\ell^{u}_{\sigma}$ satisfies

$$cl(\ell^{u}_{\sigma})\setminus(\ell^{u}_{\sigma}\cup\sigma)=\{\omega\},$$

where $\omega$ is a sink. If $\dim W^{u}_{\sigma}=1(2)$, then $cl(\ell^{u}_{\sigma})$ is the arc (the 2-sphere) topologically embedded ³ to $M^{3}$ (see, for example, Proposition 2.3 [16]). The set $\ell^{u}_{\sigma}\cup\sigma$ is the smooth submanifold of $M^{3}$, but the manifold $cl(\ell^{u}_{\sigma})$ may be wild at the point $\omega$. In this case the separatrix $\ell^{u}_{\sigma}$ is called wild, while otherwise it is called tame. The wildness and the tameness of a stable separatrix is defined in the same way.

The first “wild” example was constructed by D. Pixton in [35]. This was a diffeomorphism of Pixton class which is the class of 3-dimensional Morse – Smale diffeomorphisms whose nonwandering sets consist of exactly four points: three nodes and one saddle (Fig. 17). This class allows diffeomorphisms with wild separatrices, but either the attractor $A_{f}$ or the repeller $R_{f}$ of any diffeomorphism of Pixton class consists of the unique node. Therefore, the attractor and the repeller are separated by a 2-sphere. C. Bonatti and V. Grines [3] gave the topological classification of Pixton diffeomorphisms. Theorem 9 for Pixton diffeomorphisms was proved in [4].

Now consider a diffeomorphism $f$ constructed as the connected sum of two 2-spheres on each of which a Pixton diffeomorphism is defined (Fig. 17. The 3-balls for the connected sum are marked).

Fig. 17

The connected sum of two Pixton diffeomorphisms.

All separatrices of such a diffeomorphism are wild by construction, while the attractor $A_{f}$ (the repeller $R_{f}$) is the arc with two points of wildness. Therefore, the trapping neighborhood of the attractor is not homeomorphic to the 3-ball and the orbit space $\hat{V}_{f}$ is not homeomorphic to the manifold $S^{2}\times S^{1}$. Thus, the constructed diffeomorphism cannot be joined to the source-sink diffeomorphism by a stable arc.

3.5 The Existence of Exotic Spheres

At the end of the 1950s J. Milnor [24] proved the surprising fact that the 7-sphere allows 28 pairwise nondiffeomorphic smooth structures. One of them coincides with the standard one, i. e., the one produced on $S^{n}=\{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}:x^{2}_{1}+\dots x^{2}_{n+1}=1\}$ by two charts $(U_{-},\vartheta_{-}),(U_{+},\vartheta_{+})$ where $N(\underbrace{0,\dots,0}\limits_{n},1)$, $S(\underbrace{0,\dots,0}\limits_{n},-1)$, $U_{-}={S}^{n}\setminus\{S\}$, $U_{+}={S}^{n}\setminus\{N\}$ and $\vartheta_{-}:{S}^{n}\setminus\{S\}\to\mathbb{R}^{n}$, $\vartheta_{+}:{S}^{n}\setminus\{N\}\to\mathbb{R}^{n}$ (called stereographic projections, Fig. 18) and defined by

$$\displaystyle\vartheta_{-}(x_{1},\dots,x_{n+1})=\left(\frac{x_{1}}{1+x_{n+1}},\dots,\frac{x_{n-1}}{1+x_{n+1}},\frac{x_{n}}{1+x_{n+1}}\right),$$

$$\displaystyle\vartheta_{+}(x_{1},\dots,x_{n+1})=\left(\frac{x_{1}}{1-x_{n+1}},\dots,\frac{x_{n-1}}{1-x_{n+1}},\frac{x_{n}}{1-x_{n+1}}\right).$$

Fig. 18

The stereographic projection.

The others define the so-called exotic spheres. One of the first examples of exotic spheres found by J. Milnor [24, Chapter 3] was the following. Consider two copies of $D^{4}\times S^{3}$ with the boundary $S^{3}\times S^{3}$ and identify the point $(a,b)$ on the boundary with the point $(a,a^{2}ba^{-1})$ ⁴ . The resulting manifold has the natural smooth structure of the 7-manifold. Milnor showed that this manifold allows the Morse function with exactly two critical points, and this means that topologically it is the 7-sphere. He also showed that this manifold allows no orientation-reversing diffeomorphism, and this means that this is not the standard 7-sphere.

It was shown that the exotic structures are impossible in dimensions $1,2,3,5,6,12,56,61$. Some multidimensional spheres have exactly two differential structures, while others have thousands of them. The problem of existence of exotic 4-spheres is still unsolved.

It follows from Smale’s $h$-cobordism theorem (see, for example, [25]) that for $n>5$ each exotic $n$-sphere is diffeomorphic to the tangled sphere, i. e., the sphere obtained from two copies of the standard $n$-balls by identification of their borders by an orientation-preserving diffeomorphism of the sphere $S^{n-1}$. Though any gluing diffeomorphism is isotopic to the identity, J. Milnor showed [25] that it can be nondiffeotopic to it. Finally, it follows from the result by J. Cerf [6] that two diffeomorphisms of the sphere $S^{n-1},n>6$, define diffeomorphic smooth structures on $S^{n}$ if and only if these diffeomorphisms are diffeotopic.

Hence, nondiffeotopic orientation-preserving diffeomorphisms exist on the $n$-sphere ($n\geqslant 6$) if and only if there exist exotic $(n+1)$-spheres and the number of diffeotopy classes coincides with the number of different exotic spheres. Moreover, from Theorem 10 it follows that there is a Morse – Smale diffeomorphism in every diffeotopy class of diffeomorphisms of the $n$-sphere. Therefore, diffeomorphisms of distinct classes cannot be joined by a smooth arc, in particular, by a stable arc.

Theorem 10 ([4], Proposition 3.6).

Any orientation-preserving diffeomorphism of the $n$ -sphere is diffeotopic to a source-sink diffeomorphism.

The idea of the proof is the following. At first the initial diffeomorphism is replaced with such a diffeomorphism that the polar points of the sphere are its fixed points. Then the fixed points are turned into the hyperbolic source and the hyperbolic sink. After that the resulting map is composed with a source-sink map whose expansion at the source is great. Since all the changes are the compositions with identity diffeotopic diffeomorphisms, the resulting diffeomorphism is diffeotopic to the initial one.

4 COMPONENTS OF STABLE CONNECTEDNESS OF GRADIENT-LIKE SURFACE DIFFEOMORPHISMS

In this chapter we discuss the classification results with respect to the stable isotopy connectedness for some classes of surface gradient-like diffeomorphisms.

4.1 Gradient-like Diffeomorphisms of the 2-sphere

Consider $S^{1}$ as the equator of the sphere $S^{2}$. Then a structurally stable diffeomorphism of the circle with exactly two periodic orbits of the period $m\in\mathbb{N}$ and the rotation number $\frac{k}{m}$ can be extended to a diffeomorphism $\phi_{k,m}:S^{2}\to S^{2}$ with the two fixed sources in the north pole and in the south pole. We call this diffeomorphism the model diffeomorphism. Denote by $C_{k,m}$ the component of stable isotopy connectedness of $\phi_{k,m}$ and denote by $C^{-}_{k,m}$ the component of stable isotopy connectedness of $\phi^{-1}_{k,m}$. Denote by $C_{0}$ the component of stable isotopy connectedness of the source-sink diffeomorphism $\phi_{0}\in G$ whose nonwandering set consists of exactly one source and one sink.

The following theorem provides a complete classification of gradient-like diffeomorphisms with respect to the stable connectivity.

Theorem 11 ([32], Theorem 1.1).

Any orientation-preserving gradient-like diffeomorphism of the $2$ -sphere $S^{2}$ lies in one of the components $C_{0},C_{k,m},C^{-}_{k,m},k,m\in\mathbb{N},k<m/2,(k,m)=1$ and

• the components $C_{0},C_{k,m},C^{-}_{k,m},k,m\in\mathbb{N},k<m/2,(k,m)=1$ are pairwise disjoint;

• $C_{k,m}=C_{m-k,m},C^{-}_{k,m}=C^{-}_{m-k,m},C_{1,2}=C^{-}_{1,2}=C_{0,1}=C^{-}_{0,1}=C_{0}.$

Notice that from Theorem 6 it follows that the diffeomorphisms $\phi_{k,m},\phi_{{k^{\prime}},{m^{\prime}}}:S^{2}\to S^{2}$ for $m=2^{r}\cdot q,m^{\prime}=2^{r^{\prime}}\cdot q^{\prime}$ for integer $r,r^{\prime}\geqslant 0$ and natural $q\neq q^{\prime}$ lie in the different components of stable isotopy connectedness.

The key point of the proof of Theorem 11 is the fact that the dynamics of any orientation-preserving gradient-like diffeomorphism $f:M^{2}\to M^{2}$ can be represented as the dynamics of the global dual pair attractor-repeller whose space of the wandering orbits is connected. Denote by $\Omega^{0}_{f},\Omega^{1}_{f},\Omega^{2}_{f}$ the sets of the sinks, the saddles and the sources of $f$. For any (possibly empty) $f$-invariant set $\Sigma\subset\Omega^{1}_{f}$ let

$$A_{\Sigma}=\Omega^{0}_{f}\cup W^{u}_{\Sigma},R_{\Sigma}=\Omega^{2}_{f}\cup W^{s}_{\Omega^{1}_{f}\setminus\Sigma}.$$

From [17] it follows that $A_{\Sigma}$ and $R_{\Sigma}$ are the dual attractor and repeller. We say

$$V_{\Sigma}=M^{2}\setminus(A_{\Sigma}\cup R_{\Sigma})$$

are the characteristic space. Denote by $\hat{V}_{\Sigma}$ the orbit space of action of the diffeomorphism $f$ on $V_{\Sigma}$. According to [15], each connected component of the manifold $\hat{V}_{\Sigma}$ is homeomorphic to the 2-torus. Moreover, due to [29] there is such a set $\Sigma$ that the orbit space $\hat{V}_{\Sigma}$ is connected. Let $\Sigma$ be this set and let

$$A_{f}=A_{\Sigma},R_{f}=R_{\Sigma},V_{f}=V_{\Sigma}.$$

For the class $G$ of gradient-like diffeomorphisms on the 2-sphere $S^{2}$ the attractor $A_{f}$ and the repeller $R_{f}$ can be described more precisely (Fig. 19). Notice that $V_{f}$ consists of $m_{f}$ mutually disjoint cylinders. A collection of noncontractible curves, one curve on each component, divides the sphere $S^{2}$ into two disjoint parts $U$ and $V$ such that

$$f(U)\subset U,A_{f}=\bigcap\limits_{j\in\mathbb{N}}f^{j}(U);f^{-1}(V)\subset V,R_{f}=\bigcap\limits_{j\in\mathbb{N}}f^{-j}(V).$$

Then for any diffeomorphism $f\in G$ $($or for $f^{-1})$ the following holds:

• 1) the set $U$ consists of $m_{f}\in\mathbb{N}$ pairwise disjoint disks $D_{f},f(D_{f}),\dots,f^{m_{f}-1}(D_{f})$ such that $f^{m_{f}}(clD_{f})\subset intD_{f}$;

• 2) the attractor $A_{f}$ consists of $m_{f}$ connected components $A,f(A),\dots,f^{m_{f}-1}(A)$ such that $A=\bigcap\limits_{j\in\mathbb{N}}f^{jm_{f}}(D_{f})$ and $f^{m_{f}}(A)=A$;

• 3) the repeller $R_{f}$ is connected.

The reason for this is that the $m_{f}$ cylinders are mapped into each other by $f$ cyclically, and thus so do the boundary circles of $U$ and $V$. This implies that $U$ is either a union of $m_{f}$ disjoint discs or a connected set, and the same alternative holds for $V$. Since $U\cup V$ is a sphere, we must have that either $U$ is a union of discs and $V$ is connected, or the other way around.

Fig. 19

An example of the attractor $A_{f}$ and the repeller $R_{f}$ of a gradient-like diffeomorphism of the 2-sphere. The set $U$ is shown in rose, the part of the attractor $A_{f}$ lying in $D_{f}$ is shown in red, and the connected repeller $R_{f}$ is shown in blue.

Denote by $G^{+}$ the subset of the set $G$ consisting of diffeomorphisms with all saddle points of positive orientation type. Let $G^{-}=G\setminus G^{+}$ and denote by $G_{1}$ the subset of $G$ consisting of diffeomorphisms $f$ such that there exists a fixed pair $A_{f},R_{f}$ for which $m_{f}=1$. Due to the topology of the 2-sphere one can prove that

• $G^{-}\subset G_{1}$;

• for any diffeomorphism $f\in G^{+}$ the number $m_{f}$ is well-defined, i.e., it is independent of the choice of the pair $A_{f},R_{f}$.

Thus, the set $G^{+}\setminus G_{1}$ is the union of pairwise disjoint subsets

$$G^{+}\setminus G_{1}=G_{2}\dots\cup G_{m}\cup\dots$$

and $m_{f}=m$ for any diffeomorphism $f\in G_{m},m>1$. Since the attractor $A_{f}$ and the repeller $R_{f}$ for $f\in G_{1}$ are connected, it is possible to construct an arc joining $f$ to $\phi_{0}$. Due to connectedness of $A_{f}$ for $f\in G_{m},m>1$, it is possible to reduce $A_{f}$, i.e., to join it by a stable arc to a diffeomorphism $g$ of the class $H_{m}\subset G_{m}$ where $H_{m}$ consists of diffeomorphisms $g$ whose attractor $A_{g}$ is the unique sink orbit of period $m$ (Fig. 20).

Fig. 20

Transfer from the diffeomorphism $f\in G_{m}$ to the diffeomorphism $g\in H_{m}$. The connected component of the attractor $A_{f}$ lying in the disk $D_{f}$ is shown in red (left). It is the union of the closures of the unstable manifolds of the saddle points ${\sigma_{1}},{\sigma_{2}},{\sigma_{3}},{\sigma_{6}},{\sigma_{8}},{\sigma_{10}}$. The arc from $f$ to $g$ has six saddle-node bifurcations resulting in reduction of the attractor as shown on the right. There the attractor $A_{g}$ of the diffeomorphism $g$ consists of the sink orbit $\omega$.

Then one proves that for the diffeomorphism $g$ there exists a saddle orbit $\mathcal{O}_{\sigma}$ of period $m$ such that $clW^{u}_{\mathcal{O}_{\sigma}}$ is the $g$-invariant closed curve ${C_{\sigma}}$ and the map $g|_{C_{\sigma}}$ is topologically conjugate to the rough circle transformation with the rotation number $\frac{k}{m}$ (Fig. 21). The rotation numbers for all such circles are equal. This makes it possible to join the diffeomorphism $g$ by a stable arc to the diffeomorphism whose nonwandering set consists of one saddle orbit $\mathcal{O}_{\sigma}=\{\sigma,f(\sigma),\dots,f^{m-1}(\sigma)\}$, one sink orbit $\mathcal{O}_{\omega}=\{\omega,f(\omega),\dots,f^{m-1}(\omega)\}$ and fixed sources $\alpha_{1},\alpha_{2}$. The resulting diffeomorphism can be joined to the model diffeomorphism $\phi_{k,m}$ by a stable arc. Then one proves that distinct model diffeomorphisms cannot be joined by a stable arc.

Fig. 21

The curve $C_{\sigma}$.

4.2 Palis Diffeomorphisms

In this chapter we consider the class $P$ of orientation-preserving gradient-like diffeomorphisms $f$ on the orientable surface $M^{2}$ such that all nonwandering points of $f$ are fixed and are of positive orientation type. This class of diffeomorphisms was singled out by J. Palis in [33] as the class of Morse – Smale surface diffeomorphisms which can be included in a topological flow.

Theorem 12 ([31], Theorem 1).

Any two diffeomorphisms $f,f^{\prime}\in P$ defined on the same surface $M^{2}$ can be joined by a stable arc with a finite number of noncritical saddle-nodes which unfold generically.

First one constructs a bifurcation-free arc joining the diffeomorphism $f\in P$ to a certain diffeomorphism $\phi_{f}\in P$ which is the 1-time shift of the gradient flow of some Morse function. Then, due to Theorems 1 and 2, the diffeomorphisms $\phi_{f},\phi_{f^{\prime}}$ are joined by an arc with a finite number of saddle-node bifurcations.