An integral representation for topological pressure in terms of conditional probabilities

TL;DR

This paper presents a new integral representation of topological pressure for dynamical systems, linking it to conditional probabilities and invariant measures, with implications for approximation algorithms.

Contribution

It introduces a novel integral representation of pressure under certain conditions, extending to all invariant measures with stronger hypotheses.

Findings

Pressure can be expressed as an integral involving conditional probabilities.

The representation holds for all invariant measures under stronger assumptions.

An algorithmic method for approximating pressure is derived.

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.