Stable ergodicity of dominated systems

TL;DR

This paper introduces a new geometric approach to understanding stable ergodicity in systems with dominated splittings, highlighting that issues with Pesin's local manifolds may be less problematic than previously thought.

Contribution

It presents a novel method based on geometric analysis of stable and unstable manifolds, offering insights into non-uniform hyperbolicity.

Findings

Stable ergodicity can be analyzed through geometric properties of manifolds.

The lack of uniform size in Pesin's manifolds is less problematic than traditionally believed.

The approach provides new perspectives on non-uniform hyperbolic systems.

Abstract

We provide a new approach to stable ergodicity of systems with dominated splittings, based on a geometrical analysis of global stable and unstable manifolds of hyperbolic points. Our method suggests that the lack of uniform size of Pesin's local stable and unstable manifolds - a notorious problem in the theory of non-uniform hyperbolicity - is often less severe than it appeas to be.