The Variational Principle for a $\mathbb{Z}_+^N$ Action on a Hausdorff Locally Compact Space

TL;DR

This paper extends the concept of topological pressure to locally compact Hausdorff spaces with $ Z_+^N$ actions, establishing a variational principle linking topological and measure-theoretic pressures, and introduces admissible covers for non-compact cases.

Contribution

It introduces a new definition of topological pressure for non-compact spaces and proves a variational principle for $ Z_+^N$ actions using admissible covers.

Findings

Established a variational principle for topological pressure in non-compact spaces.

Defined topological pressure using admissible covers suitable for locally compact spaces.

Provided an example demonstrating the effectiveness of admissible covers.

Abstract

We extend the definition of topological pressure to locally compact Hausdorff spaces, and we demonstrate a "variational principle" comparing the topological and measure theoretic pressures. Given a continuous $\mathbb{Z}_+^N$-action $T$ over a locally compact Hausdorff space $X$ and a continuous function vanishing at infinity $f \in C_0(X)$, we define topological pressure $P(T,f)$ using open covers of a special type we call "admissible covers". With this topological pressure, we demonstrate that \begin{equation*} P(T,f) = \sup_{\mu} P_\mu(T,f), \end{equation*} where the supremum is taken over all $T$-invariant probability Radon measures over $X$, and is equal to $0$ when there is none. In the last section, we present an example that illustrates why admissible covers are so adequate to deal with the non-compact case, while some other approaches would fail.