Non uniformly hyperbolic dynamics: H\'enon maps and related dynamical systems

TL;DR

This paper reviews the development of non-uniform hyperbolic dynamics, focusing on Hénon maps and related systems, highlighting advances from uniform hyperbolic theory to complex chaotic behaviors in concrete dynamical models.

Contribution

It summarizes the evolution of techniques extending hyperbolic theory to non-uniformly hyperbolic systems like Hénon maps and related models.

Findings

Extension of hyperbolic theory to Hénon maps

Development of techniques for chaotic one-dimensional systems

Almost complete understanding of non-uniform hyperbolic dynamics

Abstract

In the 1960s and 1970s a large part of the theory of dynamical systems concerned the case of uniformly hyperbolic or Axiom A dynamical system and abstract ergodic theory of smooth dynamical systems. However since around 1980 an emphasize has been on concrete examples of one-dimensional dynamical systems with abundance of chaotic behavior (Collet &Eckmann and Jakobson). New proofs of Jakobson's one-dimensional results were given by Benedicks and Carleson \cite{BC85} and were considerably extended to apply to the case of H\'enon maps by the same authors \cite{BC91}. Since then there has been a considerable development of these techniques and the methods have been extended to the ergodic theory and also to other dynamical systems (work by Viana, Young, Benedicks and many others). In the cases when it applies one can now say that this theory is now almost as complete as the Axiom A theory.