Abstract
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.
Citation
Luis Barreira. Jinjun Li. Claudia Valls. "Irregular sets of two-sided Birkhoff averages and hyperbolic sets." Ark. Mat. 54 (1) 13 - 30, April 2016. https://doi.org/10.1007/s11512-015-0214-2
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