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arXiv · 1904.01045

Symplectomorphisms with positive metric entropy

Artur Avila·Sylvain Crovisier·Amie Wilkinson
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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.

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Artur Avila, Sylvain Crovisier, Amie Wilkinson. 2019-04-01. Symplectomorphisms with positive metric entropy. https://arxiv.org/abs/1904.01045

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