Arkiv för Matematik

Irregular sets of two-sided Birkhoff averages and hyperbolic sets

Luis Barreira, Jinjun Li, and Claudia Valls

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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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Ark. Mat., Volume 54, Number 1 (2016), 13-30.

Received: 10 September 2013
First available in Project Euclid: 30 January 2017

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2015 © Institut Mittag-Leffler


Barreira, Luis; Li, Jinjun; Valls, Claudia. Irregular sets of two-sided Birkhoff averages and hyperbolic sets. Ark. Mat. 54 (2016), no. 1, 13--30. doi:10.1007/s11512-015-0214-2.

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