The Structure on Invariant Measures of $C^1$ generic diffeomorphisms

TL;DR

This paper investigates the structure of invariant measures in $C^1$ generic diffeomorphisms, showing that measures supported on certain sets are characterized by orbit averages and linking hyperbolicity properties.

Contribution

It establishes a precise relationship between invariant measures and orbit averages for $C^1$ generic diffeomorphisms, and connects hyperbolicity of points with the hyperbolicity of the set.

Findings

Invariant measures coincide with accumulation measures of time averages.

Residual set of points with this property in $ ext{Lambda}$.

Non-uniform hyperbolicity of irregular points implies uniform hyperbolicity of $ ext{Lambda}$.

Abstract

Let $\Lambda$ be an isolated non-trival transitive set of a $C^1$ generic diffeomorphism $f\in\Diff(M)$. We show that the space of invariant measures supported on $\Lambda$ coincides with the space of accumulation measures of time averages on one orbit. Moreover, the set of points having this property is residual in $\Lambda$ (which implies the set of irregular$^+$ points is also residual in $\Lambda$). As an application, we show that the non-uniform hyperbolicity of irregular$^+$ points in $\Lambda$ with totally 0 measure (resp., the non-uniform hyperbolicity of a generic subset in $\Lambda$) determines the uniform hyperbolicity of $\Lambda$.