Dynamics of $C^1$-diffeomorphisms: global description and prospects for classification

TL;DR

This paper explores the structure of $C^1$-diffeomorphisms, aiming to classify their dynamics by approximating systems with hyperbolic behavior or homoclinic bifurcations, advancing understanding of their global behavior.

Contribution

It presents recent progress towards Palis' conjecture that any $C^1$-diffeomorphism can be approximated by either hyperbolic systems or those with homoclinic bifurcations.

Findings

Progress towards Palis' conjecture on $C^1$-diffeomorphisms.

Identification of dense subsets with distinct dynamical behaviors.

Insights into the global structure and classification prospects.

Abstract

We are interested in finding a dense part of the space of $C^1$-diffeomorphisms which decomposes into open subsets corresponding to different dynamical behaviors: we discuss results and questions in this direction. In particular we present recent results towards a conjecture by J. Palis: any system can be approximated either by one which is hyperbolic (and whose dynamics is well understood) or by one which exhibits a homoclinic bifurcation (a simple local configuration involving one or two periodic orbits).