Luis BarreiraDepartamento de Matemática, Instituto Superior Técnico, Universidade de LisboaJinjun LiSchool of Mathematics and Statistics, Minnan Normal UniversityClaudia VallsDepartamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa
Dynamical SystemsAlgebraic Topology and General Topologymathscidoc:1701.11013
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.