Topological Pressure for Locally Compact Metrizable Systems

TL;DR

This paper extends the concept of topological pressure and the variational principle from compact to separable locally compact metric spaces, broadening the scope of thermodynamic formalism in dynamical systems.

Contribution

It generalizes the definition of topological pressure and proves the variational principle for systems on locally compact metric spaces.

Findings

Established a new definition of topological pressure for locally compact spaces.

Proved the variational principle holds in this more general setting.

Bridged the gap between compact and non-compact dynamical systems.

Abstract

It is widely known that when $X$ is compact Hausdorff, and when $T: X \to X$ and $f: X \to \mathbb{R}$ are continuous, \begin{equation*} P(T,f) = \sup_{\text{$\mu$: Radon probability}} \left( h_\mu(T) + \int f\, \mathrm{d}\mu \right), \end{equation*} where $P(T,f)$ is the "topological pressure" and $h_\mu(T)$ is the measure theoretic entropy of $T$ with respect to $\mu$. This result is known as "variational principle". We generalize the concept of "topological pressure" for the case where $X$ is a separable locally compact metric space. Our definitions are quite similar to those used in the compact case. Our main result is the validity of the "variational principle".