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

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_μ P_μ(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.