Semicontinuity of entropy, existence of equilibrium states and   continuity of physical measures
  We obtain some results of existence and continuity of physical measures through equilibrium states and apply these to non-uniformly expanding transformations on compact manifolds with non-flat critical sets, obtaining sufficient conditions for continuity of physical measures and, for local diffeomorphisms, necessary and sufficient conditions for stochastic stability. In particular we show that, under certain conditions, stochastically robust non-uniform expansion implies existence and continuous variation of physical measures. 
Vitor Araujo