Nonuniform Center Bunching and the Genericity of Ergodicity among $C^1$ Partially Hyperbolic Symplectomorphisms

TL;DR

This paper introduces nonuniform center bunching for partially hyperbolic diffeomorphisms, extending prior results, and proves that generically, such symplectomorphisms are ergodic, also providing new examples of stably ergodic systems.

Contribution

It develops a new technique called nonuniform center bunching and proves generic ergodicity among $C^1$ partially hyperbolic symplectomorphisms, also constructing new stably ergodic examples.

Findings

$C^1$-generic partially hyperbolic symplectomorphisms are ergodic

Introduction of nonuniform center bunching technique

Construction of new stably ergodic diffeomorphisms

Abstract

We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns--Wilkinson and Avila--Santamaria--Viana. Combining this new technique with other constructions, we prove that $C^1$-generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms.