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

Abstract

Let $Λ$ 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 $Λ$ coincides with the space of accumulation measures of time averages on one orbit. Moreover, the set of points having this property is residual in $Λ$ (which implies the set of irregular$^+$ points is also residual in $Λ$). As an application, we show that the non-uniform hyperbolicity of irregular$^+$ points in $Λ$ with totally 0 measure (resp., the non-uniform hyperbolicity of a generic subset in $Λ$) determines the uniform hyperbolicity of $Λ$.