Upper semi-continuity of entropy map for non-uniformly hyperbolic  systems

Abstract: For a continuous transformation f on a compact manifold M, the entropy map of f is defined by the metric entropy on the set of all f-invariant measures and it is generally not continuous. However, it is still worth our effort to investigate the upper semi-continuity of since, for instance, it implies the existence of invariant measures of maximal entropy. In this talk I will talk about the upper semi-continuity of entropy map for non-uniformly hyperbolic systems. We prove that for C1 non-uniformly hyperbolic systems with domination, the entropy map is upper semi-continuous; then we extend result to the interval map, and give upper semi-continuity of entropy map for C^{1+\alpha} interval maps  from the view point of folding entropy.