Irregular sets of two-sided Birkhoff averages and hyperbolic sets

Luis Barreira Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa Jinjun Li School of Mathematics and Statistics, Minnan Normal University Claudia Valls Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa

Dynamical Systems Algebraic Topology and General Topology mathscidoc:1701.11013

Arkiv for Matematik, 54, (1), 13-30, 2013.9
For two-sided topological Markov chains, we show that the set of points for which the two-sided Birkhoff averages of a continuous function diverge is residual. We also show that the set of points for which the Birkhoff averages have a given set of accumulation points other than a singleton is residual. A nontrivial consequence of our results is that the set of points for which the local entropies of an invariant measure on a locally maximal hyperbolic set does not exist is residual. This strongly contrasts to the Shannon–McMillan–Breiman theorem in the context of ergodic theory, which says that local entropies exist on a full measure set.
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@inproceedings{luis2013irregular,
  title={Irregular sets of two-sided Birkhoff averages and hyperbolic sets},
  author={Luis Barreira, Jinjun Li, and Claudia Valls},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203641322167089},
  booktitle={Arkiv for Matematik},
  volume={54},
  number={1},
  pages={13-30},
  year={2013},
}
Luis Barreira, Jinjun Li, and Claudia Valls. Irregular sets of two-sided Birkhoff averages and hyperbolic sets. 2013. Vol. 54. In Arkiv for Matematik. pp.13-30. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203641322167089.
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