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

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

2010-04-20

arxiv.org/abs/1004.3439v1

Acta Mathematica Sinica, English Series, 2012 Volume 28, Number 4, 817-824

Mathematics

Dynamical Systems

Scientific paper

10.1007/s10114-011-9723-5

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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