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, realvalued function f on X, we show that the corresponding local pressure P (T, f ;U) satisfies P (T, f ;U) = sup{hμ(T,U) + ∫ X f(x)dμ(x) : μ is a T -invariant measure}, and 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 hμ(T,U) = inf f∈C(X,R) {P (T, f ;U)− ∫