An integral representation for topological pressure in terms of conditional probabilities

Abstract

Given an equilibrium state $\mu$ for a continuous function $f$ on a shift of finite type $X$, the pressure of $f$ is the integral, with respect to $\mu$, of the sum of $f$ and the information function of $\mu$. We show that under certain assumptions on $f$, $X$ and an invariant measure $\nu$, the pressure of $f$ can also be represented as the integral with respect to $\nu$ of the same integrand. Under stronger hypotheses we show that this representation holds for all invariant measures $\nu$. We establish an algorithmic implication for approximation of pressure, and we relate our results to a result in thermodynamic formalism.