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Hyperbolic Invariant Sets With Positive Measures

Dynamical Systems 2007-05-23 v2

Abstract

If a C1+aC^{1 + a}, a>0a >0, volume-preserving diffeomorphism on a compact manifold has a hyperbolic invariant set with positive volume, then the map is Anosov. We also give a direct proof of ergodicity of volume-preserving CC1+aCC^{1+a}, a>0a>0, Anosov diffeomorphism without the usual Hopf argument.

Keywords

Cite

@article{arxiv.math/0503437,
  title  = {Hyperbolic Invariant Sets With Positive Measures},
  author = {Zhihong Xia},
  journal= {arXiv preprint arXiv:math/0503437},
  year   = {2007}
}