Smale regular and chaotic A-homeomorphisms and A-diffeomorphisms

TL;DR

This paper introduces a hierarchy of Smale A-homeomorphisms on manifolds, classifies their invariant sets, and provides conditions for conjugacy, with applications to surface diffeomorphisms and Morse-Smale systems.

Contribution

It defines and analyzes Smale A-homeomorphisms, including regular, semi-chaotic, chaotic, and super chaotic types, and establishes criteria for their conjugacy and classification.

Findings

Invariant sets determine the dynamics of various A-homeomorphisms.

Necessary and sufficient conditions for conjugacy of these systems are established.

Complete classification of Morse-Smale diffeomorphisms on certain manifolds is achieved.

Abstract

We introduce Smale A-homeomorphisms that includes regular, semi-chaotic, chaotic, and super chaotic homeomorphisms of topo\-lo\-gi\-cal $n$-manifold $M^n$, $n\geq 2$. Smale A-homeo\-mor\-p\-hisms contain A-diffeomorphisms (in particular, structurally stable diffeomorphisms) provided $M^n$ admits a smooth structure. Regular A-homeomorphisms contain all Morse-Smale diffeomorphisms, while semi-chaotic and chaotic A-homeomorphisms contain A-diffeo\-mor\-p\-hisms with trivial and nontrivial basic sets. Super chaotic A-homeo\-mor\-p\-hisms contain A-diffeomorphisms whose basic sets are nontrivial. We describe invariant sets that determine completely dynamics of regular, semi-chaotic, and chaotic Smale A-homeo\-mor\-p\-hisms. This allows us to get necessary and sufficient conditions of conjugacy for these Smale A-homeomorp\-hisms. We apply this necessary and sufficient conditions for structu\-ral\-ly stable surface diffeomorphisms with arbitrary number of one-dimensional expanding attractors. We also use this conditions to get the complete classification of Morse-Smale diffeomorphisms on projective-like $n$-manifolds for $n=2,8,16$.