Variational principles of pressure

Abstract

Given a topological dynamical system $(X, T)$, a Borel cover$\mathcal{U}$ of $X$ and a sub-additive sequence $\mathcal{F}$ ofreal-valued continuous functions on $X$, two notions ofmeasure-theoretical pressure $P_\mu^- (T, \mathcal{U}, \mathcal{F})$and $P_\mu^+ (T, \mathcal{U}, \mathcal{F})$ for an invariant Borelprobability measure $\mu$ are introduced. When $\mathcal{U}$ is anopen cover, a local variational principle between topological andmeasure-theoretical pressure is proved; it is also established theupper semi-continuity of P•+$(T, \mathcal{U}, \mathcal{F})$and P•+$(T, \mathcal{U}, \mathcal{F})$ on the space of allinvariant Borel probability measures. The notions ofmeasure-theoretical pressure $P_\mu^- (T, X, \mathcal{F})$ and$P_\mu^+ (T, X, \mathcal{F})$ for an invariant Borel probabilitymeasure $\mu$ are also introduced. A global variational principlebetween topological and measure-theoretical pressure is alsoobtained.