Perturbations of C1-diffeomorphisms and dynamics of generic conservative diffeomorphisms of surface

TL;DR

This paper surveys properties of C1-generic conservative surface diffeomorphisms, highlighting their transitivity, and discusses recent perturbation techniques and connecting lemmas that underpin these dynamical behaviors.

Contribution

It provides a survey of known properties and recent developments in C1-perturbation techniques for conservative surface diffeomorphisms, emphasizing transitivity results.

Findings

C1-generic conservative diffeomorphisms of surfaces are transitive

Connecting lemmas for pseudo-orbits are central to these results

Recent perturbation methods facilitate the proof of transitivity

Abstract

In the first part of this text we give a survey of the properties satisfied by the C1-generic conservative diffeomorphisms of compact surfaces. The main result that we will discuss is that a C1-generic conservative diffeomorphism of a connected compact surface is transitive. It is obtain as a consequence of a connecting lemma for pseudo-orbits. In the last parts we expose some recent developments of the C1-perturbation technics and the proof of this connecting lemma. We are not aimed to deal with technicalities nor to give the finest available versions of these results. Besides this theory exists also in higher dimension and in the non-conservative setting, we restricted the scope of this presentation to the conservative case on surfaces, since it offers some simplifications which allow to explain in an easier way the main ideas of the subject.