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E. Zhuzhoma

Publications and source records attributed to E. Zhuzhoma.

12 recordsLinked to original sources
arXiv2025

Nonsingular structural stable chaotic 3-flows of attractor-repeller type

We show that any orientable closed 3-manifold $M$ admits structurally stable non-singular flow $f^t$ whose non-wandering set $NW(f^t)$ consists of a 2-dimensional expanding attractor and finitely many repelling periodic trajectories. For $M=\mathbb{S}^3$, we prove that the set of repelling periodic trajectories can be an arbitrary link provided that this link contains the figure eight knot. When a link consists of a unique repelling periodic trajectory (not necessarily a figure eight knot), we prove that this trajectory cannot be a torus knot. For any closed 3-manifold $M$, we show that there does not admit any structurally stable non-singular flow $f^t$ whose non-wandering set $NW(f^t)$ consists of a 2-dimensional expanding attractor and a repelling periodic trajectory so that the repelling periodic trajectory is a trivial knot (i.e., it bounds a disk in $M$).

math.DS↗
arXiv2024

On existence of expanding attractors with different dimensions

We prove that $n$-sphere $\mathbb{S}^n$, $n\geq 2$, admits structurally stable diffeomorphisms $\mathbb{S}^n\to\mathbb{S}^n$ with non-orientable expanding attractors of any topological dimension $d\in\{1,\ldots,[\frac{n}{2}]\}$ where $[x]$ is an integer part of $x$. One proves that $n$-torus $\mathbb{T}^n$, $n\geq 2$, admits structurally stable diffeomorphisms $\mathbb{T}^n\to\mathbb{T}^n$ with orientable expanding attractors of any topological dimension $1\leq q\leq n-1$. We also prove that given any closed $n$-manifold $M^n$, $n\geq 2$, 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 non-orientable expanding attractor. Similar statements hold for axiom A flows.

math.DS↗
arXiv2024

On a classification of axiom A diffeomorphisms with codimension one basic sets and isolated saddles

Let $M^n$, $n\geq 3$, be a closed orientable $n$-manifold and $\mathbb{D}_k(M^n;a,b,c)$ the set of axiom A diffeomorp\-hisms $f: M^n\to M^n$ satisfying the following conditions: (1) $f$ has $k\geq 1$ nontrivial basic sets each is either an orientable codimension one expanding attractor or an orientable codimension one contracting repeller, and other trivial basic sets which are $a$ sinks, $b$ sources, $c$ saddles; (2) the invariant manifolds of isolated saddles are intersected transversally. We classify the diffeomorphisms from $\mathbb{D}_k(M^n;a,b,c)$ up to the global conjugacy on non-wandering sets for the following subsets $\mathbb{S}_k(M^n;a,b,c), \mathbb{P}_k(M^n;0,0,1), \mathbb{M}_k(M^n;0,0,1)$ of $\mathbb{D}_k(M^n;a,b,c)$ where $\mathbb{S}_k(M^n;a,b,c)$ satisfies to the following conditions: ($1_{\mathbb{S}}$) every nontrivial basic set of any $f\in\mathbb{S}_k(M^n;a,b,c)$ is uniquely bunched, and there is at least one nontrivial attractor and at least one nontrivial repeller, i.e. $k\geq 2$; ($2_{\mathbb{S}}$) $c\geq 1$ and all isolated saddles have the same Morse index belonging to $\{1,n-1\}$. The subset $\mathbb{P}_k(M^n;0,0,1)\subset\mathbb{D}_k(M^n;0,0,1)$ satisfies to the following conditions: ($1_{\mathbb{P}}$) any boundary point of $f\in\mathbb{P}_k(M^n;0,0,1)$ is fixed; ($2_{\mathbb{P}}$) a unique isolated saddle has Morse index different from $\{1,n-1\}$. The subset $\mathbb{M}_k(M^n;0,0,1)\subset\mathbb{D}_k(M^n;0,0,1)$ satisfies to the following conditions: ($1_{\mathbb{M}}$) any boundary point of $f\in\mathbb{M}_k(M^n;0,0,1)$ is fixed; ($2_{\mathbb{M}}$) a unique isolated saddle has Morse index belonging to $\{1,n-1\}$. The classification is based on a description of topological structure of supporting manifolds $M^n$.

math.DS↗
arXiv2019

On 2-dimensional expanding attractors of A-flows

We prove that given any closed $n$-manifold $M^n$, $n\geq 4$, there is an A-flow $f^t$ on $M^n$ such that the non-wandering set $NW(f^t)$ consists of 2-dimensional expanding attractor (the both, orientable and non-orientable) and trivial basic sets. For 3-manifolds, we prove that given any closed 3-manifold $M^3$, there is an A-flow $f^t$ on $M^3$ such that the non-wandering set $NW(f^t)$ consists of a non-orientable 2-dimensional expanding attractor and trivial basic sets. Moreover, there is a nonsingular A-flow $f^t$ on a 3-sphere such that the non-wandering set $NW(f^t)$ consists of an orientable 2-dimensional expanding attractor and trivial basic sets (isolated periodic trajectories).

math.DS↗
arXiv2018

On conjugacy of Smale homeomorphisms

Given closed topological $n$-manifold $M^n$, $n\geq 2$, one introduces the classes of Smale regular $SRH(M^n)$ and Smale semi-regular $SsRH(M^n)$ homeomorphisms of $M^n$ with $SRH(M^n)\subset~SsRH(M^n)$. The class $SRH(M^n)$ contains all Morse-Smale diffeomorphisms, while $SsRH(M^n)$ contains A-diffeomorphisms with trivial and some nontrivial basic sets provided $M^n$ admits a smooth structure. We select invariant sets that determine dynamics of Smale homeomorphisms. This allows us to get necessary and sufficient conditions of conjugacy for $SRH(M^n)$ and $SsRH(M^n)$. We deduce applications for some Morse-Smale diffeomorphisms and A-diffeomorphisms with codimension one expanding attractors.

math.DS↗
arXiv2016

On spectral decomposition of Smale-Vietoris axiom A diffeomorphisms

We introduce Smale-Vietoris diffeomorphisms that include the classical DE-mappings with Smale solenoids. We describe the correspondence between basic sets of axiom A Smale-Vietoris diffeomorphisms and basic sets of nonsingular axiom A endomorphisms. For Smale-Vietoris diffeomorphisms of 3-manifolds, we prove the uniqueness of nontrivial solenoidal basic set. We construct a bifurcation between different types of solenoidal basic sets which can be considered as a destruction (or birth) of Smale solenoid.

math.DS↗
arXiv2015

On heteroclinic separators of magnetic fields in electrically conducting fluids

In this paper we partly solve the problem of existence of separators of a magnetic field in plasma. We single out in plasma a 3-body with a boundary in which the movement of plasma is of special kind which we call an (a-d)-motion. We prove that if the body is the 3-annulus or the "fat" orientable surface with two holes the magnetic field necessarily have a heteroclinic separator. The statement of the problem and the suggested method for its solution lead to some theoretical problems from Dynamical Systems Theory which are of interest of their own.

physics.plasm-ph↗
arXiv2004

Proof of the Morse conjecture for analytic flows on orientable surfaces

In 1946, M. Morse proposed a conjecture that an analytic topologically transitive systems is metrically transitive. We prove this Morse conjecture for flows on a closed orientable surface of negative Euler characteristic. As a consequence, the Morse conjecture is true for highly transitive flows on non-orientable surfaces.

math.DS↗
arXiv2004

On arithmetical and dynamical properties of Lorentz maps of the torus

We proved that any Lorentz transformation of 2-torus is Anosov automorphism. One completely describes admissible parameters of Lorentz transformations and their arithmetical properties. One proved that an admissible speed light parameter has a countable spectra accumulating to this parameter.

math.DS↗