Evgeniy Zhuzhoma
ul. Bolshaya Pecherskaya 25/12, Nizhni Novgorod, 603005, Russia
National research University “Higher school of Economics”
Publications:
Barinova M. K., Grines V. Z., Pochinka O. V., Zhuzhoma E. V.
Hyperbolic Attractors Which are Anosov Tori
2024, vol. 29, no. 2, pp. 369375
Abstract
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) $\dim{\Lambda}=n1$, $\dim{W^u_x}=1$, $x\in\Lambda$;
2) $\dim\,\Lambda=k,\,\dim\, W^u_x=k1,\,x\in\Lambda$, and $\Lambda$ belongs to an $f$invariant closed $k$manifold, $2\leqslant k\leqslant n$, topologically embedded in $M^n$.

Grines V. Z., Medvedev V. S., Zhuzhoma E. V.
Classification of Axiom A Diffeomorphisms with Orientable Codimension One Expanding Attractors and Contracting Repellers
2024, vol. 29, no. 1, pp. 143155
Abstract
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 $M^n$.

Medvedev V. S., Zhuzhoma E. V.
Smale Regular and Chaotic AHomeomorphisms and ADiffeomorphisms
2023, vol. 28, no. 2, pp. 131147
Abstract
We introduce Smale Ahomeomorphisms that include regular, semichaotic, chaotic,
and superchaotic homeomorphisms of a topological $n$manifold $M^n$, $n\geqslant 2$. Smale Ahomeomorphisms
contain axiom A diffeomorphisms (in short, Adiffeomorphisms) provided that $M^n$
admits a smooth structure. Regular Ahomeomorphisms contain all Morse–Smale diffeomorphisms,
while semichaotic and chaotic Ahomeomorphisms contain Adiffeomorphisms with
trivial and nontrivial basic sets. Superchaotic Ahomeomorphisms contain Adiffeomorphisms
whose basic sets are nontrivial. The reason to consider Smale Ahomeomorphisms instead of
Adiffeomorphisms may be attributed to the fact that it is a good weakening of nonuniform
hyperbolicity and pseudohyperbolicity, 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 Ahomeomorphisms. This allows us to get necessary and sufficient conditions of conjugacy for these Smale Ahomeomorphisms (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. 
Grines V. Z., Medvedev V. S., Zhuzhoma E. V.
On the Topological Structure of Manifolds Supporting Axiom A Systems
2022, vol. 27, no. 6, pp. 613628
Abstract
Let $M^n$, $n\geqslant 3$, be a closed orientable $n$manifold and $\mathbb{G}(M^n)$ the set of Adiffeomorp\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 onedimensional 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 $\mathbb S^{n1}\times \mathbb S^1$ 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 $n1$, $0\leqslant l_f\leqslant \lambda_f$, $\lambda_f\geqslant 0$ is the number of all periodic points whose Morse index does not belong to the set $\{0,1,n1,n\}$ of diffeomorphism $f$. Similar statements hold for gradientlike 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.

Grines V. Z., Zhuzhoma E. V.
Expanding attractors
2006, vol. 11, no. 2, pp. 225246
Abstract
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; DAattractor; Plykin attractor; Robinson–Williams attractors), one reviews the Williams theory, Bothe's classification of onedimensional solenoids in 3manifolds, Grines–Plykin–Zhirov's classification of onedimensional 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
