On hyperbolic attractors and repellers of endomorphisms

TL;DR

This paper investigates hyperbolic attractors and repellers in endomorphisms, providing criteria for their identification and analyzing dynamics on codimension-one basic sets, including smooth embedding and expansion properties.

Contribution

It introduces a criterion for basic sets of A-endomorphisms to be attractors and studies the dynamics on codimension-one basic sets, including smooth embedding and expansion conditions.

Findings

Attractors of codimension one are smoothly embedded and expanding.

Basic sets of type (n,0) are repellers with expanding dynamics.

Criteria for identifying attractors among basic sets are established.

Abstract

It is well known that topological classification of dynamical systems with hyperbolic dynamics is significantly defined by dynamics on nonwandering set. F. Przytycki generalized axiom $A$ for smooth endomorphisms that was previously introduced by S. Smale for diffeomorphisms and proved spectral decomposition theorem which claims that nonwandering set of an $A$-endomorphism is a union of a finite number basic sets. In present paper the criterion for a basic sets of an $A$-endomorphism to be an attractor is given. Moreover, dynamics on basic sets of codimension one is studied. It is shown, that if an attractor is a topological submanifold of codimension one of type $(n-1, 1)$, then it is smoothly embedded in ambient manifold and restriction of the endomorphism to this basic set is an expanding endomorphism. If a basic set of type $(n, 0)$ is a topological submanifold of codimension one, then it is a repeller and restriction of the endomorphism to this basic set is also an expanding endomorphism.