On decomposition of ambient surfaces admitting A-diffeomorphisms with nontrivial attractors and repellers

TL;DR

This paper explores the relationship between A-diffeomorphisms with multiple basic sets on surfaces and the topology of the ambient surface, revealing how the surface's structure is determined by the dynamics of these sets.

Contribution

It characterizes the topology of surfaces supporting A-diffeomorphisms with multiple basic sets and shows these diffeomorphisms are Omega-stable but not structurally stable.

Findings

Ambient surface is a connected sum of surfaces and tori.

Genus of each surface depends on dynamical properties.

Diffeomorphisms are Omega-stable but not structurally stable.

Abstract

It is well-known that there is a close relationship between the dynamics of diffeomorphisms satisfying the axiom $A$ and the topology of the ambient manifold. In the given article, this statement is considered for the class $\mathbb G(M^2)$ of $A$-diffeomorphisms of closed orientable surfaces such that their non-wandering set consists of $k_f\geq 2$ connected components of one-dimensional basic sets (attractors and repellers). We prove that the ambient surface of every diffeomorphism $f\in \mathbb G(M^2)$ is homeomorphic to the connected sum of $k_f$ closed orientable surfaces and $l_f$ two-dimensional tori such that the genus of each surface is determined by the dynamical properties of appropriating connected component of a basic set and $l_f$ is determined by the number and position of bunches, belonging to all connected components of basic sets. We also prove that every diffeomorphism from the class $\mathbb G(M^2)$ is $\Omega$-stable but is not structurally stable.