Dominated Pesin theory: convex sum of hyperbolic measures

TL;DR

This paper explores the relationship between hyperbolic measures and ergodic measures in non-uniformly hyperbolic systems, establishing conditions under which measures can be approximated by hyperbolic ergodic measures sharing a common intersection class.

Contribution

It introduces the concept of intersection classes for hyperbolic measures with dominated splittings and characterizes when measures are approximated by hyperbolic ergodic measures with the same intersection class.

Findings

Measures with dominated splitting are approximated by hyperbolic ergodic measures sharing the same intersection class.

The importance of the domination assumption is demonstrated through examples.

A criterion is provided for when a measure is accumulated by ergodic measures with the same hyperbolic index.

Abstract

In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures? To every hyperbolic measure $\mu$ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class $H(\mu)$ of periodic orbits for the homoclinic relation, called its \emph{intersection class}. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index such as the dominated splitting is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class. We provide examples which indicate the importance of the domination assumption.