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".
Categories: math.DS