# American Institute of Mathematical Sciences

November  2009, 24(4): 1409-1435. doi: 10.3934/dcds.2009.24.1409

## Variational principles of pressure

 1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  May 2008 Revised  December 2008 Published  May 2009

Given a topological dynamical system $(X, T)$, a Borel cover $\mathcal{U}$ of $X$ and a sub-additive sequence $\mathcal{F}$ of real-valued continuous functions on $X$, two notions of measure-theoretical pressure $P_\mu^- (T, \mathcal{U}, \mathcal{F})$ and $P_\mu^+ (T, \mathcal{U}, \mathcal{F})$ for an invariant Borel probability measure $\mu$ are introduced. When $\mathcal{U}$ is an open cover, a local variational principle between topological and measure-theoretical pressure is proved; it is also established the upper semi-continuity of P+$(T, \mathcal{U}, \mathcal{F})$ and P+$(T, \mathcal{U}, \mathcal{F})$ on the space of all invariant Borel probability measures. The notions of measure-theoretical pressure $P_\mu^- (T, X, \mathcal{F})$ and $P_\mu^+ (T, X, \mathcal{F})$ for an invariant Borel probability measure $\mu$ are also introduced. A global variational principle between topological and measure-theoretical pressure is also obtained.
Citation: Guohua Zhang. Variational principles of pressure. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1409-1435. doi: 10.3934/dcds.2009.24.1409
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