# Symplectomorphisms with positive metric entropy

## TL;DR

This paper establishes a dichotomy for generic symplectomorphisms, showing they either have zero Lyapunov exponents almost everywhere or are partially hyperbolic and ergodic, advancing the understanding of symplectic dynamics.

## Contribution

It proves a dichotomy for $C^1$-generic symplectomorphisms, completing a program by Ricardo Mañé and generalizing key results on accessibility.

## Key findings

- Almost all points have zero Lyapunov exponents or the map is ergodic and partially hyperbolic.
- Stable accessibility is dense among partially hyperbolic diffeomorphisms.
- The result extends to partially hyperbolic invariant sets.

## Abstract

We obtain a dichotomy for $C^1$-generic symplectomorphisms: either all the Lyapunov exponents of almost every point vanish, or the map is partially hyperbolic and ergodic with respect to volume. This completes a program first put forth by Ricardo Ma\~n\'e. A main ingredient in our proof is a generalization to partially hyperbolic invariant sets of the main result in [Dolgopyat-Wilkinson] that stable accessibility is $C^1$ dense among partially hyperbolic diffeomorphisms.