Ergodic Properties of Invariant Measures for C^{1+\alpha} nonuniformly hyperbolic systems

TL;DR

This paper extends Sigmund's results to non-uniformly hyperbolic systems, showing that invariant measures can be approximated by time averages of single-orbit Dirac measures, with irregular points forming a residual set.

Contribution

It generalizes the ergodic properties of invariant measures from uniformly to non-uniformly hyperbolic systems, highlighting the density and residuality of certain orbit-based measures.

Findings

Invariant measures are approximated by single-orbit time averages.

Irregular points form a residual subset of the support.

Results extend Sigmund's uniform hyperbolic case to non-uniform hyperbolic systems.

Abstract

For an ergodic hyperbolic measure $\omega$ of a $C^{1+{\alpha}}$ diffeomorphism, there is an $\omega$ full-measured set $\tilde\Lambda$ such that every nonempty, compact and connected subset $V$ of $\mathbb{M}_{inv}(\tilde\Lambda)$ coincides with the accumulating set of time averages of Dirac measures supported at {\it one orbit}, where $\mathbb{M}_{inv}(\tilde\Lambda)$ denotes the space of invariant measures supported on $\tilde\Lambda$. Such state points corresponding to a fixed $V$ are dense in the support $supp(\omega)$. Moreover, $\mathbb{M}_{inv}(\tilde\Lambda)$ can be accumulated by time averages of Dirac measures supported at {\it one orbit}, and such state points form a residual subset of $supp(\omega)$. These extend results of Sigmund [9] from uniformly hyperbolic case to non-uniformly hyperbolic case. As a corollary, irregular points form a residual set of $supp(\omega)$.