Smale regular and chaotic A-homeomorphisms and A-diffeomorphisms

Diffeomorphisms satisfying Smale’s axiom A (in short, A-diffeomorphisms) were introduced by Smale [44] as a magnificent and natural generalization of structurally stable diffeomorphisms. By definition, a non-wandering set of A-diffeomorphism has a uniform hyperbolic structure and is the topological closure of periodic orbits. Smale proved that the non-wandering set splits into closed, transitive, and invariant pieces called basic sets. A basic set is trivial, if it is an isolated periodic orbit. A good example of A-diffeomorphism with trivial basic sets is a Morse-Smale diffeomorphism [36, 43]. Such diffeomorphisms demonstrate regular dynamics. Due to Bowen [9], A-diffeomorphisms with nontrivial basic sets demonstrate chaotic dynamics since any such diffeomorphism has a positive entropy. The most familiar nontrivial basic sets are Plykin’s attractor [37] and codimension one expanding attractors introduced by Williams [46, 47]. Such basic sets appeared in various applications, see for example [15, 25, 45].

Taking in mind that there are manifolds that do not admit smooth structures [33], we introduce Smale A-homeomorphisms with non-wandering sets having a hyperbolic type (see a precise definition below). Such homeomorphisms naturally appear in topological dynamical systems. For example, in [11], it was proved the existence of topological Morse functions with three critical points on topological (including nonsmoothable) closed manifolds. Starting with these examples, one can construct topological (maybe, only topological) Morse-Smale flows and Morse-Smale homeomorphisms with the non-wandering set consisting of three fixed points of hyperbolic type. Deep theory of topological dynamical systems was developed in [1, 2].

The challenging problem in the Theory of Dynamical Systems is the classification up to conjugacy dynamical systems with regular and chaotic dynamics. Recall that homeomorphisms $f_{1}$, $f_{2}:M^{n}\to M^{n}$ are called conjugate, if there is a homeomorphism $h:M^{n}\to M^{n}$ such that $h\circ f_{1}=f_{2}\circ h$. To check whether given $f_{1}$ and $f_{2}$ are conjugate, one constructs usually an invariant of conjugacy which is a dynamical characteristic keeping under a conjugacy homeomorphism. Normally, such invariant is constructed in the frame of special class of dynamical systems. The famous invariant is Poincare’s rotation number for the class of transitive circle homeomorphisms [38]. This invariant is effective i.e. two transitive circle homeomorphisms are conjugate if and only if they have the same Poincare’s rotation number (see [35] and [5], ch. 7, concerning invariants of low dimensional dynamical systems). Anosov [3] and Smale [44] were first who realize the fundamental role of hyperbolicity for dynamical systems. Numerous topological invariants were constructed for special classes of A-diffeomorphisms including Anosov systems [12, 29, 34] and Morse-Smale systems, see the books [6, 13] and the surveys [14, 30].

In the frame of Smale A-homeomorphisms, we introduce regular, semi-chaotic, chaotic, and super chaotic homeomorphisms. We get necessary and sufficient conditions of conjugacy for regular, semi-chaotic, and chaotic Smale A-homeomorphisms on a closed topological $n$-manifold $M^{n}$, $n\geq 2$. Automatically, this gives necessary and sufficient conditions of conjugacy for Morse-Smale diffeomorphisms and a wide class of A-diffeomorphisms with nontrivial basic sets provided $M^{n}$ admits a smooth structure. We apply our conditions for structurally stable surface diffeomorphisms with arbitrary number of one-dimensional expanding attractors. We classify Morse-Smale diffeomorphisms with three periodic points on high-dimensional projective-like manifolds. Note that the projective-like manifolds were introduced by the authors in [32] (see also [31]).

Let us give the main definitions and formulate the main results. Later on, $clos\,N$ means the topological closure of $N$. In [32], the authors introduced the notation of equivalent embedding as follows. Let $M^{k}_{1}$, $M^{k}_{2}\subset M^{n}$ be topologically embedded $k$-manifolds, $1\leq k\leq n-1$. We say they have the equivalent embedding if there are neighborhoods $U(clos\,M^{k}_{1})$, $U(clos\,M^{k}_{2})$ of $clos\,M^{k}_{1}$, $clos\,M^{k}_{2}$ respectively and a homeomorphism $h:U(clos\,M^{k}_{1})\to U(clos\,M^{k}_{2})$ such that $h(M^{k}_{1})=M^{k}_{2}$. This notation allows to classify Morse-Smale topological flows with non-wandering sets consisting of three equilibriums [32]. To be precise, it was proved that two such flows $f^{t}_{1}$, $f^{t}_{2}$ are topologically equivalent if and only if the stable (or unstable) separatrices of saddles of $f^{t}_{1}$, $f^{t}_{2}$ have the equivalent embedding. Remark that the notation of equivalent embedding goes back to a scheme introduced by Leontovich and Maier [26, 27] to attack the classification problem for flows on 2-sphere.

Solving the conjugacy problem for homeomorphisms, we have to add conjugacy relations to the equivalent embedding. The modification of (global) conjugacy is a local conjugacy when the conjugacy holds in some neighborhoods of compact invariant sets. We introduce the intermediate notion, so-called a locally equivalent dynamical embedding (in short, dynamical embedding), as follows. Let $f_{1}$, $f_{2}:M^{n}\to M^{n}$ be homeomorphisms of closed topological $n$-manifold $M^{n}$, $n\geq 2$, and $N_{1}$, $N_{2}$ invariant sets of $f_{1}$, $f_{2}$ respectively i.e. $f_{i}(N_{i})=N_{i}$, $i=1,2$. We say that the sets $N_{1}$, $N_{2}$ have the same dynamical embedding if there are neighborhoods $\delta_{1}$, $\delta_{2}$ of $clos\leavevmode\nobreak\ N_{1}$, $clos\leavevmode\nobreak\ N_{2}$ respectively and a homeomorphism $h_{0}:\delta_{1}\cup f_{1}(\delta_{1})\to M^{n}$ such that

Recall that $F:L^{n}\to L^{n}$ is an A-diffeomorphism of smooth manifold $L^{n}$ provided the non-wandering set $NW(F)$ is hyperbolic, and the periodic orbits of $F$ are dense in $NW(F)$ [44]. The hyperbolicity implies that every point $z_{0}\in NW(F)$ has the stable $W^{s}(z_{0})$ and unstable $W^{u}(z_{0})$ manifolds formed by points $y\in L^{n}$ such that $\varrho_{L}(F^{pk}z_{0},F^{pk}y)\to 0$ as $k\to+\infty$ and $k\to-\infty$ respectively, where $\varrho_{L}$ is a metric on $L^{n}$ [18, 19, 21, 23, 39, 44]. Moreover, $W^{s}(z_{0})$ and $W^{u}(z_{0})$ are homeomorphic (in the interior topology) to Euclidean spaces $\mathbb{R}^{\dim W^{s}(z_{0})}$, $\mathbb{R}^{\dim W^{u}(z_{0})}$ respectively. Note that $\dim W^{s}(z_{0})+\dim W^{u}(z_{0})=n$. The non-wandering set $NW(F)$ is a finite union of pairwise disjoint $F$-invariant closed sets $\Omega_{1}$, $\ldots,\Omega_{k}$ such that every restriction $F|_{\Omega_{i}}$ is topologically transitive. These $\Omega_{i}$ are called basic sets of $F$. A basic set is nontrivial if it is not a periodic isolated orbit. Set $W^{s(u)}(\Omega_{i})=\cup_{x\in\Omega_{i}}W^{s(u)}(x)$. One says that $\Omega_{i}$ is a sink (source) basic set provided $W^{u}(\Omega_{i})=\Omega_{i}$ ($W^{s}(\Omega_{i})=\Omega_{i}$). A basic set $\Omega_{i}$ is a saddle basic set if it is neither a sink nor a source basic set.

A homeomorphism $f:M^{n}\to M^{n}$ is called a Smale A-homeomorphism if there is an A-diffeomorphism $F:L^{n}\to L^{n}$ such that the non-wandering sets $NW(f)$, $NW(F)$ have the same dynamical embedding. As a consequence, $NW(f)$ is a finite union of pairwise disjoint $f$-invariant closed sets $\Lambda_{1}$, $\ldots,\Lambda_{k}$ called basic sets of $f$ such that every restriction $f|_{\Lambda_{i}}$ is topologically transitive. Each basic set $\Lambda$ has the stable manifold $W^{s}(\Lambda)$, and the unstable manifold $W^{u}(\Lambda)$. Similarly, one introduces the families of sink basic sets $\omega(f)$, and source basic sets $\alpha(f)$, and saddle basic sets $\sigma(f)$.

A Smale A-homeomorphism $f$ is called regular if all basic sets $\omega(f)$, $\sigma(f)$, $\alpha(f)$ are trivial.

A Smale A-homeomorphism $f$ is called semi-chaotic if exactly one family from the families $\omega(f)$, $\sigma(f)$, $\alpha(f)$ consists of non-trivial basic sets.

A Smale A-homeomorphism $f$ is called chaotic if exactly two families from the families $\omega(f)$, $\sigma(f)$, $\alpha(f)$ consists of non-trivial basic sets.

A Smale A-homeomorphism $f$ is called super chaotic if the families $\omega(f)$, $\sigma(f)$, $\alpha(f)$ consists of non-trivial basic sets.

In Section 1, we represent examples of all types above of Smale A-homeomorphisms. Actually, all examples are A-diffeomorphisms.

Now let us introduce invariant sets that determine dynamics of Smale homeomorphisms. Given any Smale A-homeomorphism $f:M^{n}\to M^{n}$, denote by $A(f)$ (resp., $R(f)$) the union of $\omega(f)$ (resp., $\alpha(f)$) and unstable (resp., stable) manifolds of saddle basic sets $\sigma(f)$ :

The following statement gives the necessary and sufficient conditions of conjugacy for three types of Smale A-homeomorphisms. Note that if $f$ is a regular or semi-chaotic Smale A-homeomorphism, then at least one of the families $\omega(f)$, $\alpha(f)$ consists of trivial basic sets. However, if $f$ is a chaotic Smale A-homeomorphism, then it is possible a priori that the both $\omega(f)$ and $\alpha(f)$ consist of nontrivial basic sets but $\sigma(f)$ consists of trivial basic sets.

In Section 4, we apply Theorem 1 to consider the conjugacy for structurally stable surface diffeomorphisms $M^{2}\to M^{2}$ with one-dimensional (orientable and non-orientable) attractors $\Lambda_{1}$, $\ldots$, $\Lambda_{k}$, $k\geq 2$ (remark that a structurally stable diffeomorphism is an A-diffeomorphism [28]). We also use Theorem 1 to classify Morse-Smale diffeomorphisms with three periodic points on projective-like $n$-manifolds for $n\in\{2,8,16\}$.

First, we prove the following statement interesting itself (note that a one dimensional expanding attractor is a trivial basic set).

The case $k=s_{0}=1$ is represented in Fig. 1, (a), while the case $k=2$ and $s_{0}=1$ is represented in Fig. 1, (b) with two Plykin attractors. See also [7] where one got the estimate for the number of one-dimensional basic sets of surface A-diffeomorphisms depending on a genus of supporting surface.

The following statement shows that the dynamical embedding of unstable manifolds of isolated saddles (trivial basic sets) determine completely global dynamics of structurally stable surface diffeomorphisms with one-dimensional expanding attractors.

Denote by $SRH(M^{n})$ the class of Smale regular homeomorphisms $M^{n}\to M^{n}$. Note that it is possible that $f\in SRH(M^{n})$ has the empty set $\sigma(f)$ of saddle periodic points. In this case the set $\alpha(f)$ consists of a unique source and the set $\omega(f)$ consists of a unique sink, and $M^{n}=S^{n}$ is an $n$-sphere. Later on, we’ll assume that $f\in SRH(M^{n})$ has a non-empty set $\sigma(f)$ of saddle periodic points.

Clearly, $SRH(M^{n})$ contains all Morse-Smale diffeomorphisms provided $M^{n}$ admits a smooth structure. Note that the class $SRH(M^{n})$ is an essential extension of Morse-Smale diffeomorphisms for a Smale regular diffeomorphism can contain nonhyperbolic periodic points, tangencies, and separatrix connections.

As a consequence of Theorem 1, one gets the following statement (in particular, one gets the necessary and sufficient conditions of conjugacy for any Morse-Smale diffeomorphisms on smooth closed manifolds).

Denote by $MS(M^{n};a,b,c)$ the set of Morse-Smale diffeomorphisms $f:M^{n}\to M^{n}$ whose non-wandering set consists of $a$ sinks, $b$ sources, and $c$ saddles. In [31], the authors proved that for $MS(M^{n};1,1,1)$ the only values of $n$ possible are $n\in\{2,4,8,16\}$. Moreover, the supporting manifolds for $MS(M^{n};1,1,1)$ are projective-like provided $n\in\{2,8,16\}$ [31, 32].

First, to illustrate the applicability of Corollary 1, we consider very simple class $MS(M^{2};1,1,1)$. In this case, the supporting manifold $M^{2}$ is the projective plane $M^{2}=\mathbb{P}^{2}$ [31]. Below, we define a type for a unique saddle of $f\in MS(\mathbb{P}^{2},1,1,1)$. Using Corollary 1 we’ll show how to get the following complete classification of Morse-Smale diffeomorphisms $MS(\mathbb{P}^{2},1,1,1)$.

Thus, up to conjugacy, there are four classes of Morse-Smale diffeomorphisms $MS(\mathbb{P}^{2},1,1,1)$.

The most essential application is a complete classification of Morse-Smale diffeomorphisms $MS(M^{8};1,1,1)$ and $MS(M^{16};1,1,1)$. The supporting $2k$-manifolds for diffeomorphisms from the set $MS^{2k}(1,1,1)$ will be denoted by $M^{2k}(1,1,1)$. Recall that $S^{k}$ is a $k$-sphere. Below, $\alpha_{f}$, $\sigma_{f}$, and $\omega_{f}$ mean the source, the saddle, and the sink of $f\in MS^{2k}(1,1,1)$ respectively.

An embedding $\varphi:S^{k}\to M^{2k}(1,1,1)$ is called basic if

• •

  $\varphi(S^{k})$ is a locally flat $k$-sphere;

• •

  $M^{2k}(1,1,1)\setminus\varphi(S^{k})$ is an open $2k$-ball, $M^{2k}(1,1,1)=B^{2k}\sqcup\varphi(S^{k})$.

It was proved in [31] that every supporting manifold $M^{2k}(1,1,1)$, $k=4,8$, admits a basic embedding.

Thus, every $f\in MS^{2k}(1,1,1)$ corresponds the basic embedding $\varphi(f):S^{k}\to M^{2k}(1,1,1)$. Given any basic embedding $\varphi$, there is $f\in MS^{2k}(1,1,1)$ such that $\varphi(f)=\varphi$. At last, a dynamical embedding of basic embedding defines completely a conjugacy class in $MS^{2k}(1,1,1)$. We see, that the set of basic embedding (up to isotopy) form the admissible set of conjugacy invariants for the Morse-Smale diffeomorphisms $MS^{2k}(1,1,1)$, $k=4,8$. As to the class $MS(M^{4};1,1,1)$, the existence of realizable and effective conjugacy invariant is still the open problem. The main reason is the possibility of wild embedding for topological closure of separatrix of a saddle [31].

The next statement shows that an equivalent embedding gives an invariant of conjugacy for iterations of Morse-Smale diffeomorphisms $f\in MS^{2k}(1,1,1)$, $k=4,8$.

The structure of the paper is the following. In Section 2, we give some previous results. In Section 3, we prove Theorem 1. In Section 4, we prove Propositions 1, 2, and Theorems 2, 3, 4.

Acknowledgments. This work is supported by Laboratory of Dynamical Systems and Applications of National Research University Higher School of Economics, of the Ministry of science and higher education of the RF, grant ag. № 075-15-2019-1931.

1) Regular A-diffeomorphisms. Obvious example of regular A-diffeomorphism is a Morse-Smale diffeomorphism. Note that there are regular A-diffeomorphisms that do not belong to the set of Morse-Smale diffeomorphisms. For example, they can belong to the boundary of the set of Morse-Smale diffeomorphisms in the space of diffeomorphisms. There are regular A-diffeomorphisms which can not be approximated by Morse-Smale diffeomorphisms [40].

2) Semi-chaotic A-diffeomorphisms. A good example of semi-chaotic diffeomorphism is a so-called DA-diffeomorphism obtained from Anosov automorphism after Smale surgery [44], see Fig. 2.

A classical DA-diffeomorphism $f:\mathbb{T}^{2}\to\mathbb{T}^{2}$ contains a nontrivial attractor $\omega(f)$, trivial repeller $\alpha(f)$, and empty set $\sigma(f)$. A generalized DA-diffeomorphism contains nonempty set $\sigma(f)$ [17]. Another example of semi-chaotic A-diffeomorphism is a classical Smale horseshoe $g_{s}:\mathbb{S}^{2}\to\mathbb{S}^{2}$. Well-known that there is $g_{s}$ with trivial attractor $\omega(g_{s})$ and repeller $\alpha(g_{s})$, and nontrivial $\sigma(g_{s})$.

Starting with DA-diffeomorphisms, Williams [46] constructed an open domain $\mathcal{N}\subset Diff^{1}(\mathbb{T}^{2})$ consisting of structurally unstable diffeomorphisms. It is easy to see that $\mathcal{N}$ contains semi-chaotic A-diffeomorphisms.

One more example of semi-chaotic A-diffeomorphism is shown in Fig. 3 with a DA-attractor and Plykin attractor on a torus.

3) Chaotic A-diffeomorphisms. Take the classical DA-diffeomorphism $f:\mathbb{T}^{2}\to\mathbb{T}^{2}$ with the non-wandering set consisting of a source $\alpha$ and one-dimensional expanding attractor $\Lambda_{a}$. The diffeomorphism $f^{-1}$ defined on a copy $\mathbb{T}^{2}$ has the non-wandering set consisting of a sink $\omega$ and one-dimensional contracting repeller $\Lambda_{r}$. Let us delete a small neighborhood $U_{a}$ (resp., $U_{s}$) of $\alpha$ (resp., $\omega$) homeomorphic to a disk. Take an orientation reversing diffeomorphism $h:\partial U_{a}\to\partial U_{r}$. Then the surface $M^{2}=\left(\mathbb{T}^{2}\setminus U_{a}\right)\cup_{h}\left(\mathbb{T}^{2}\setminus U_{r}\right)$ is a pretzel (closed orientable surface of genus 2). Following [42], one can construct A-diffeomorphism $g:M^{2}\to M^{2}$ with the non-wandering set consisting of $\Lambda_{a}\cup\Lambda_{r}$ such that $g|_{\Lambda_{a}}=f$ and $g|_{\Lambda_{r}}=f^{-1}$. Thus, $\alpha(g)=\Lambda_{r}$ and $\omega(g)=\Lambda_{a}$. Clearly, $g$ is a chaotic A-diffeomorphism. Due to [42], there is a construction such that $g$ has a closed simple curve consisting of the tangencies of the invariant stable manifolds of $\Lambda_{a}$ and the invariant unstable manifolds of $\Lambda_{r}$.

Another example one gets starting with a Smale solenoid [44], see Fig. 2. This mapping can be extended to $\Omega$-stable diffeomorphism $f_{s}:M^{3}\to M^{3}$ with a one-dimensional expanding attractor, say $\Omega_{1}$, and one-dimensional contracting repeller, say $\Omega_{2}$, where $M^{3}$ is a 3-sphere or lens space [8, 24]. This chaotic diffeomorphism is similar to the Robinson-Williams diffeomorphism $g$ considered above. There is a bifurcation of $\Omega_{1}$ into a zero-dimensional saddle type basic set and isolated attracting periodic orbits [48]. As a result, one gets a chaotic Smale diffeomorphism $f_{0}:M^{3}\to M^{3}$ with trivial basic sets $\omega(f_{0})$, and the nontrivial source basic set $\alpha(f_{0})=\Omega_{2}$, and the nontrivial zero-dimensional saddle basic set $\sigma(f_{0})$.

4) Super chaotic A-diffeomorphisms. Let $g_{s}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ be the classical Smale horseshoe and $f:\mathbb{T}^{2}\to\mathbb{T}^{2}$ the classical DA-diffeomorphism considered above. Delete small neighborhoods $U_{1}$, $U_{2}$ of the sink $\omega(g_{s})$ and the source $\alpha(g_{s})$ respectively each homeomorphic to a disk. There are reversing orientation diffeomorphisms $h_{1}:\partial U_{1}\to\partial U_{a}$ and $h_{2}:\partial U_{2}\to\partial U_{r}$. Then the surface $M^{2}=\left(\mathbb{T}^{2}\setminus U_{a}\right)\bigcup_{h_{1}}\left(S^{2}\setminus U_{1}\cup U_{2}\right)\bigcup_{h_{2}}\left(\mathbb{T}^{2}\setminus U_{r}\right)$ is a pretzel. Similarly to Robinson-Williams’s method developed in [42], one can construct a diffeomorphism $g_{0}:M^{2}\to M^{2}$ with $\alpha(g_{0})=\Lambda_{r}$, $\omega(g_{0})=\Lambda_{a}$, and $\sigma(g_{0})$ homeomorphic to the Smale horseshoe $\sigma(g_{s})$. Thus, $g_{0}$ is a super chaotic A-diffeomorphism. By similar way, one can get another examples starting with semi-chaotic A-diffeomorphisms.

We begin by recalling several definitions. Further details may be found in [5, 6, 44]. Denote by $Orb(x)$ the orbit of point $x\in M^{n}$ under a homeomorphism $f:M^{n}\to M^{n}$. The $\omega$-limit set $\omega(x)$ of the point $x$ consists of the points $y\in M^{n}$ such that $f^{k_{i}}(x)\to y$ for some sequence $k_{i}\to\infty$. Clearly that any points of $Orb(x)$ have the same $\omega$-limit. Replacing $f$ with $f^{-1}$, one gets an $\alpha$-limit set. Obviously, $\omega(x)\cup\alpha(x)\subset NW(f)$ for every $x\in M^{n}$.

Later on, $f\in SsH(M^{n})$. Given a family $C=\{c_{1},\ldots,c_{l}\}$ of sets $c_{i}\subset M^{n}$, denote by $\widetilde{C}$ the union $c_{1}\cup\ldots\cup c_{l}$. It follows immediately from definitions that

Proof. Suppose that $\omega(x)\subset\widetilde{\sigma(f)}$. Since $\widetilde{\alpha(f)}$ and $\widetilde{\omega(f)}$ are invariant sets, $x\notin\widetilde{\alpha(f)}\cup\widetilde{\omega(f)}$. Therefore, there are exist a neighborhood $U(\alpha)$ of $\alpha(f)$ and neighborhood $U(\omega)$ of $\omega(f)$ such that the positive semi-orbit $Orb^{+}(x)$ belongs to the compact set $N=M^{n}\setminus\left(U(\omega)\cup U(\alpha)\right)$. Let $V(\sigma_{1})$, $\ldots$, $V(\sigma_{m})$ be pairwise disjoint neighborhoods of saddle basic sets $\sigma_{1}$, $\ldots$, $\sigma_{m}$ respectively such that $\cup_{i=1}^{m}V(\sigma_{i})\subset N$. Since every $V(\sigma_{i})$ does not intersect $\cup_{j\neq i}V(\sigma_{j})$ and all saddle basic sets are invariant, one can take the neighborhoods $V(\sigma_{1})$, $\ldots$, $V(\sigma_{m})$ so small that every $f(V(\sigma_{i}))$ does not intersect $\cup_{j\neq i}V(\sigma_{j})$. Suppose the contrary, i.e. there is no a unique saddle basic set $\sigma_{*}\in\sigma(f)$ with $x\in W^{s}(\sigma_{*})$. Thus, there are at least two different saddle basic sets $\sigma_{1}$, $\sigma_{2}$ such that $x\in W^{s}(\sigma_{1})$ and $x\in W^{s}(\sigma_{2})$. Hence, $\omega(x)$ have to intersect $\sigma_{1}$, $\sigma_{2}$. It follows that the compact set $N_{0}=N\setminus\left(\cup_{i=1}^{m}V(\sigma_{i})\right)$ contains infinitely many points of the semi-orbit $Orb^{+}(x)$. This implies $\omega(x)\cap N_{0}\neq\emptyset$ that contradicts (2). The second assertion is proved similarly. $\Box$

A set $U$ is a trapping region for $f$ if $f\left(clos\leavevmode\nobreak\ U\right)\subset int\leavevmode\nobreak\ U.$ A set $A$ is an attracting set for $f$ if there exists a trapping set $U$ such that

A set $A^{*}$ is a repelling set for $f$ if there exists a trapping region $U$ for $f$ such that

Another words, $A^{*}$ is an attracting set for $f^{-1}$ with the trapping region $M^{n}\setminus U$ for $f^{-1}$. When we wish to emphasize the dependence of an attracting set $A$ or a repelling set $A^{*}$ on the trapping region $U$ from which it arises, we denote it by $A_{U}$ or $A^{*}_{U}$ respectively.

Let $A$ be an attracting set for $f$. The basin $B(A)$ of $A$ is the union of all open trapping regions $U$ for $f$ such that $A_{U}=A$. One can similarly define the notion of basin for a repelling set.

Let $N$ be an attracting or repelling set and $B(N)$ the basin of $N$. A closed set $G(N)\subset B(N)\setminus N$ is called a generating set for the domain $B(N)\setminus N$ if

Moreover,

1) every orbit from $B(N)\setminus N$ intersects $G(N)$; 2) if an orbit from $B(N)\setminus N$ intersects the interior of $G(N)$, then this orbit intersects $G(N)$ at a unique point; 3) if an orbit from $B(N)\setminus N$ intersects the boundary of $G(N)$, then the intersection of this orbit with $G(N)$ consists of two points; 4) the boundary of $G(N)$ is the union of finitely many compact codimension one topological submanifolds.

Proof. It is enough to prove the first statement only. Since all basic sets $\alpha(f)$ are trivial and consists of locally hyperbolic source periodic points, there is a trapping region $T(\alpha)$ for $f^{-1}$ of the set $\widetilde{\alpha(f)}$ consisting of pairwise disjoint open $n$-balls $b_{1}$, $\ldots$, $b_{r}$ such that each $b_{i}$ contains a unique periodic point $q_{i}$ from $\alpha(f)$ [36, 43]. Thus,

As a consequence, there is the generating set $G(\alpha)=\cup_{i=1}^{r}\left(clos\leavevmode\nobreak\ f^{p_{i}}(b_{i})\setminus b_{i}\right)$ consisting of pairwise disjoint closed $n$-annuluses $a_{i}=clos\leavevmode\nobreak\ f^{p_{i}}(b_{i})\setminus b_{i}$, $i=1,\ldots,r$.

Since the balls $b_{1}$, $\ldots$, $b_{r}$ are pairwise disjoint and $clos\leavevmode\nobreak\ b_{i}\subset f^{p_{i}}(b_{i})$, the balls $f^{p_{1}}(b_{1})$, $\ldots$, $f^{p_{r}}(b_{r})$ are pairwise disjoint also. For simplicity of exposition, we’ll assume that $\alpha(f)$ consists of fixed points (otherwise, $\alpha(f)$ is divided into periodic orbits each considered like a point). Therefore,

Hence, $M^{n}\setminus\cup_{i=1}^{r}b_{i}$ is a trapping region for $f$. Clearly, $A(f)\subset M^{n}\setminus\cup_{i=1}^{r}b_{i}$.

Take a point $x\in M^{n}\setminus\cup_{i=1}^{r}b_{i}$. Obviously, $\omega(x)\notin\widetilde{\alpha(f)}$. It follows from (2) that $\omega(x)\in\widetilde{\omega(f)}\cup\widetilde{\sigma(f)}$. By Lemma 1, $\omega(x)\in A(f)$. Therefore, $A(f)$ is an attracting set with the trapping region $M^{n}\setminus\cup_{i=1}^{r}b_{i}$ for $f$ :