Evgeniy Zhuzhoma

We study Axiom A diffeomorphisms (in the sense of Smale) on a closed orientable $n$-manifold $M^n$, $n\geqslant 4$, whose nonwandering sets consist of codimension one orientable attractors, repellers, and isolated saddle periodic orbits with the same Morse index. Such systems combine chaotic dynamics on basic sets with regular dynamics near trivial periodic orbits. We prove that the supporting manifold $M^n$ is homeomorphic to a connected sum of several copies of the $n$-dimensional torus. Moreover, we obtain a complete classification of such systems up to global conjugacy on their nonwandering sets. The invariant of conjugacy is a finite collection of Anosov automorphisms of the torus, called a tube, defined up to equivalence under linear automorphisms with determinants of the same sign. This extends the classical Plykin – Grines classification of individual codimension one basic sets to systems with multiple coexisting attractors and repellers.

We prove that an $n$-sphere $\mathbb{S}^n$, $n\geqslant 2$, admits structurally stable diffeomorphisms $\mathbb{S}^n\to\mathbb{S}^n$ with nonorientable expanding attractors of any topological dimension $d\in\{1,\ldots,[\frac{n}{2}]\}$ where $[x]$ is the integer part of $x$. In addition, any $n$-sphere $\mathbb{S}^n$, $n\geqslant 3$, admits axiom A diffeomorphisms $\mathbb{S}^n\to\mathbb{S}^n$ with orientable expanding attractors of any topological dimension $d\in\{1,\ldots,[\frac{n}{3}]\}$. We prove that an $n$-torus $\mathbb{T}^n$, $n\geqslant 2$, admits structurally stable diffeomorphisms $\mathbb{T}^n\to\mathbb{T}^n$ with orientable expanding attractors of any topological dimension $d\in\{1,\ldots,n-1\}$. We also prove that, given any closed $n$-manifold $M^n$, $n\geqslant 2$, and any $d\in\{1,\ldots,[\frac{n}{2}]\}$, there is an axiom A diffeomorphism $f: M^n\to M^n$ with a $d$-dimensional nonorientable expanding attractor. Similar statements hold for axiom A flows.

We consider a topologically mixing hyperbolic attractor $\Lambda\subset M^n$ for a diffeomorphism $f:M^n\to M^n$ of a compact orientable $n$-manifold $M^n$, $n>3$. Such an attractor $\Lambda$ is called an Anosov torus provided the restriction $f|_{\Lambda}$ is conjugate to Anosov algebraic automorphism of $k$-dimensional torus $\mathbb T^k$. We prove that $\Lambda$ is an Anosov torus for two cases: 1) $\dim{\Lambda}=n-1$, $\dim{W^u_x}=1$, $x\in\Lambda$; 2) $\dim\,\Lambda=k,\,\dim\, W^u_x=k-1,\,x\in\Lambda$, and $\Lambda$ belongs to an $f$-invariant closed $k$-manifold, $2\leqslant k\leqslant n$, topologically embedded in $M^n$.

Let $\mathbb{G}_k^{cod 1}(M^n)$, $k\geqslant 1$, be the set of axiom A diffeomorphisms such that the nonwandering set of any $f\in\mathbb{G}_k^{cod 1}(M^n)$ consists of $k$ orientable connected codimension one expanding attractors and contracting repellers where $M^n$ is a closed orientable $n$-manifold, $n\geqslant 3$. We classify the diffeomorphisms from $\mathbb{G}_k^{cod 1}(M^n)$ up to the global conjugacy on nonwandering sets. In addition, we show that any $f\in\mathbb{G}_k^{cod 1}(M^n)$ is $\Omega$-stable and is not structurally stable. One describes the topological structure of a supporting manifold $M^n$.

We introduce Smale A-homeomorphisms that include regular, semichaotic, chaotic, and superchaotic homeomorphisms of a topological $n$-manifold $M^n$, $n\geqslant 2$. Smale A-homeomorphisms contain axiom A diffeomorphisms (in short, A-diffeomorphisms) provided that $M^n$ admits a smooth structure. Regular A-homeomorphisms contain all Morse–Smale diffeomorphisms, while semichaotic and chaotic A-homeomorphisms contain A-diffeomorphisms with trivial and nontrivial basic sets. Superchaotic A-homeomorphisms contain A-diffeomorphisms whose basic sets are nontrivial. The reason to consider Smale A-homeomorphisms instead of A-diffeomorphisms may be attributed to the fact that it is a good weakening of nonuniform hyperbolicity and pseudo-hyperbolicity, a subject which has already seen an immense number of applications.
We describe invariant sets that determine completely the dynamics of regular, semichaotic, and chaotic Smale A-homeomorphisms. This allows us to get necessary and sufficient conditions of conjugacy for these Smale A-homeomorphisms (in particular, for all Morse–Smale diffeomorphisms). We apply these necessary and sufficient conditions for structurally stable surface diffeomorphisms with an arbitrary number of expanding attractors. We also use these conditions to obtain a complete classification of Morse–Smale diffeomorphisms on projectivelike manifolds.

Let $M^n$, $n\geqslant 3$, be a closed orientable $n$-manifold and $\mathbb{G}(M^n)$ the set of A-diffeomorp\-hisms $f: M^n\to M^n$ whose nonwandering set satisfies the following conditions: $(1)$ each nontrivial basic set of the nonwandering set is either an orientable codimension one expanding attractor or an orientable codimension one contracting repeller; $(2)$ the invariant manifolds of isolated saddle periodic points intersect transversally and codimension one separatrices of such points can intersect only one-dimensional separatrices of other isolated periodic orbits. We prove that the ambient manifold $M^n$ is homeomorphic to either the sphere $\mathbb S^n$ or the connected sum of $k_f \geqslant 0$ copies of the torus $\mathbb T^n$, $\eta_f\geqslant 0$ copies of $\mathbb S^{n-1}\times \mathbb S^1$ and $l_f\geqslant 0$ simply connected manifolds $N^n_1, \dots, N^n_{l_f}$ which are not homeomorphic to the sphere. Here $k_f\geqslant 0$ is the number of connected components of all nontrivial basic sets, $\eta_{f}=\frac{\kappa_f}{2} -k_f+\frac{\nu_f - \mu_f +2}{2},$ $ \kappa_f\geqslant 0$ is the number of bunches of all nontrivial basic sets, $\mu_f\geqslant 0$ is the number of sinks and sources, $\nu_f\geqslant 0$ is the number of isolated saddle periodic points with Morse index $1$ or $n-1$, $0\leqslant l_f\leqslant \lambda_f$, $\lambda_f\geqslant 0$ is the number of all periodic points whose Morse index does not belong to the set $\{0,1,n-1,n\}$ of diffeomorphism $f$. Similar statements hold for gradient-like flows on $M^n$. In this case there are no nontrivial basic sets in the nonwandering set of a flow. As an application, we get sufficient conditions for the existence of heteroclinic intersections and periodic trajectories for Morse – Smale flows.

The article is a survey on local and global structures (including classification results) of expanding attractors of diffeomorphisms $f : M \to M$ of a closed smooth manifold $M$. Beginning with the most familiar expanding attractors (Smale solenoid; DA-attractor; Plykin attractor; Robinson–Williams attractors), one reviews the Williams theory, Bothe's classification of one-dimensional solenoids in 3-manifolds, Grines–Plykin–Zhirov's classification of one-dimensional expanding attractors on surfaces, and Grines–Zhuzhoma's classification of codimension one expanding attractors of structurally stable diffeomorphisms. The main theorems are endowed with ideas of proof