A local variational principle of pressure and its applications to equilibrium states

We prove a local variational principle of pressure for any given open cover. More precisely, for a given dynamical system (X, T), an open cover of X, and a continuous, real-valued function f on X, we show that the corresponding local pressure P(T, f; ) satisfies , moreover, the supremum can be attained by a T-invariant ergodic measure. By establishing the upper semi-continuity and affinity of the entropy map relative to an open cover, we further show that for any T-invariant measure μ of (X, T), i.e., local pressures determine local measure-theoretic entropies. As applications, properties of both local and global equilibrium states for a continuous, real-valued function are studied. The first author is partially supported by NSFC Grants 10531010 and 10401031, program of new century excellent talents in universities, special foundation on excellent Ph.D thesis, and presidential award of the Chinese Academy of Sciences. The second author is partially supported by NSF grant DMS0204119 and NSFC grant 10428101.