Some properties of surface diffeomorphisms

TL;DR

This paper studies generic surface diffeomorphisms, establishing key properties like finiteness of attractors and hyperbolic classes, and proving equivalences among various hyperbolicity conditions, thus advancing the understanding of surface dynamics.

Contribution

It provides new results on the structure of generic surface diffeomorphisms, including finiteness of attractors and equivalences between hyperbolicity conditions, settling a notable conjecture.

Findings

Finiteness of non-trivial attractors for generic surface diffeomorphisms

Equivalence between essential hyperbolicity and hyperbolicity of dissipative homoclinic classes

Finiteness of sinks is equivalent to finiteness of spiral sinks

Abstract

We obtain some properties of $C^1$ generic surface diffeomorphisms as finiteness of {\em non-trivial} attractors, approximation by diffeomorphisms with only a finite number of {\em hyperbolic} homoclinic classes, equivalence between essential hyperbolicity and the hyperbolicity of all {\em dissipative} homoclinic classes (and the finiteness of spiral sinks). In particular, we obtain the equivalence between finiteness of sinks and finiteness of spiral sinks, abscence of domination in the set of accumulation points of the sinks, and the equivalence between Axiom A and the hyperbolicity of all homoclinic classes. These results improve \cite{A}, \cite{a}, \cite{m} and settle a conjecture by Abdenur, Bonatti, Crovisier and D\'{i}az \cite{abcd}.