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

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 $\Lambda$.

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