Topological Pressure for Locally Compact Metrizable Systems

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{$μ$: Radon probability}} \left( h_μ(T) + \int f\, \mathrm{d}μ\right), \end{equation*} where $P(T,f)$ is the "topological pressure" and $h_μ(T)$ is the measure theoretic entropy of $T$ with respect to $μ$. 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".

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