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Some systems with $C^{1}$ regularity and only negative Lyapunov exponents
Congcong Qu
Keywords:Negative Lyapunov exponents, skew product systems, periodic point.
Abstract:
      In this paper, we prove that for a $C^{1}$ diffeomorphism preserving an ergodic measure $\mu$ with only negative Lyapunov exponents, the support set of $\mu$ is a periodic orbit. For a skew product system preserving an ergodic measure with only negative fiberwise exponents, whose fiber maps are $C^{1}$ diffeomorphisms, we get that for almost all the points the disintegration of this measure is supported on finitely many points.