Evgeniy Zhuzhoma
ul. Bolshaya Pecherskaya 25/12, Nizhni Novgorod, 603005, Russia
National research University “Higher school of Economics”
Publications:
Medvedev V. S., Zhuzhoma E. V.
On a Classification of Chaotic Laminations which are Nontrivial Basic Sets of Axiom A Flows
2023, Vol. 19, no. 2, pp. 227237
Abstract
We prove that, given any $n\geqslant 3$ and $2\leqslant q\leqslant n1$, there is a closed $n$manifold $M^n$ admitting a chaotic lamination of codimension $q$ whose support has the topological dimension ${nq+1}$. For $n=3$ and $q=2$, such chaotic laminations can be represented as nontrivial 2dimensional basic sets of axiom A flows on 3manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repellerattractor dynamics. For the first type of casing, the isolated periodic trajectories form a fibered link. The second type of casing is a locally trivial fiber bundle over a circle. In the latter case, we classify (up to neighborhood equivalence) such nonmixing basic sets on their casings with solvable fundamental groups. To be precise, we reduce the classification of basic sets to the classification (up to neighborhood conjugacy) of surface diffeomorphisms with onedimensional basic sets obtained previously by V. Grines, R. Plykin and Yu. Zhirov [16, 28, 31].

Grines V. Z., Zhuzhoma E. V.
Cantor Type Basic Sets of Surface $A$endomorphisms
2021, Vol. 17, no. 3, pp. 335345
Abstract
The paper is devoted to an investigation of the genus of an orientable closed surface $M^2$
which admits $A$endomorphisms whose nonwandering set contains a onedimensional strictly
invariant contracting repeller $\Lambda_r^{}$ with a uniquely defined unstable bundle and with
an admissible boundary of finite type. First, we prove that, if $M^2$ is a torus or a
sphere, then $M^2$ admits such an endomorphism. We also show that, if $ \Omega$ is a basic set with a uniquely defined unstable bundle of the endomorphism $f\colon M^2\to M^2$ of a closed orientable surface $M^2$ and $f$ is not a diffeomorphism, then $ \Omega$ cannot be a Cantor type expanding attractor. At last, we prove that, if $f\colon M^2\to M^2$ is an $A$endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a onedimensional strictly invariant contracting repeller of Cantor type $\Omega_r^{}$ with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of $\Omega_r^{}$ is regular, then $M^2$ is a twodimensional torus $\mathbb{T}^2$ or a twodimensional sphere $\mathbb{S}^2$.

Zhuzhoma E. V., Medvedev V. S., Isaenkova N. V.
On the topological structure of the magnetic field of regions of the photosphere
2017, Vol. 13, No. 3, pp. 399412
Abstract
In this paper, using methods of Morse – Smale dynamical systems, we consider the topological
structure of the magnetic field of regions of the photosphere for a pointcharge model. For an
arbitrary number of charges (regardless of their location), without assuming a potentiality of
the field $\boldsymbol{\vec B}$ (and hence without applying specific formulas), we give estimates that connect the
numbers of charges of a certain type with the numbers of nullpoints. For the boundary estimates,
we describe the topological structure of the magnetic field. We present a bifurcation of the birth
of a large number of separators.

Grines V. Z., Gurevich E. Y., Zhuzhoma E. V., Zinina S. K.
Heteroclinic curves of Morse – Smale cascades and separators in magnetic field of plasma
2014, Vol. 10, No. 4, pp. 427438
Abstract
We obtain properties of threedimensional phase space and dynamics of Morse–Smale diffeomorphism that led to existence of at least one heteroclinical curve in nonwandering set of the diffeomorphism. We apply this result to solve a problem of existence of separators in magnetic field of plasma.
