Diffeomorphisms with positive metric entropy

TL;DR

This paper establishes a dichotomy for generic volume-preserving diffeomorphisms, showing they are either non-hyperbolic with zero Lyapunov exponents or ergodic and non-uniformly hyperbolic, advancing a program by Ricardo Mañé.

Contribution

It proves a dichotomy for $C^1$-generic volume-preserving diffeomorphisms, completing a long-standing program in dynamical systems theory.

Findings

Almost all points have zero Lyapunov exponents or the system is ergodic and non-uniformly hyperbolic.

The dichotomy characterizes the typical behavior of volume-preserving diffeomorphisms.

The result confirms a conjecture related to the structure of generic dynamical systems.

Abstract

We obtain a dichotomy for $C^1$-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Ma\~n\'e.