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 U of X, and a continuous, real-valued function f on X, we show that the corresponding local pressure P(T, f;U) satisfies P(T, f;U)={h(mu)(T,U) + integral(x) f(x)d mu(x) : mu is a T-invariant measure}, moreover, the spectrum 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 [GRAPHICS] for any T-invariant mu of (X,T), i,e., local pressure determine local measure-theoretic entropies. As applications, properties of both local and global equilibrium states for a continuous, real-valued function are studied.