Evgeniy Zhuzhoma
ul. Bolshaya Pecherskaya 25/12, Nizhni Novgorod, 603005, Russia
National research University “Higher school of Economics”
Publications:
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
