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.
We will prove Sarnak’s conjecture on Möbius disjointness for continuous interval maps of zero entropy and also for orientation-preserving circle homeomorphisms by reducing these result to a well-known theorem of Davenport from 1937.
Andrey GogolevDepartment of Mathematical Sciences, Binghamton UniversityPedro OntanedaDepartment of Mathematical Sciences, Binghamton UniversityFederico Rodriguez HertzMathematics Department, Pennsylvania State University
We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms. Laying aside many surgery constructions of 3-dimensional Anosov flows, these are the first new examples of manifolds which admit partially hyperbolic diffeomorphisms in the past forty years.