Entropy and Its Variational Principle for Locally Compact Metrizable Systems

TL;DR

This paper extends the variational principle for topological entropy to all continuous maps on locally compact metrizable systems, and applies these results to linear transformations, showing they have zero entropy.

Contribution

It generalizes the variational principle for topological entropy to non-proper maps on locally compact metrizable spaces, expanding previous results.

Findings

Extended the variational principle to all continuous maps.

Proved linear transformations on finite-dimensional vector spaces have zero entropy.

Applied the results to continuous endomorphisms of connected Lie groups.

Abstract

For a given topological dynamical system $(X,T)$ over a compact set $X$ with a metric $d$, the "variational principle" states that \begin{equation*} \sup_{\mu}h_\mu(T) = h(T) = h_d(T), \end{equation*} where $h_\mu(T)$ is the Kolmogorov-Sinai entropy, with the supremum taken over every $T$-invariant probability measure, $h_d(T)$ is the Bowen entropy, and $h(T)$ is the topological entropy as defined by Adler, Konheim and McAndrew. In [9], the concept of topological entropy was adapted for the case where $T$ is a proper map and $X$ is locally compact separable and metrizable, and the variational principle was extended to \begin{equation*} \sup_{\mu}h_\mu(T) = h(T) = \min_d h_d(T), \end{equation*} where the minimum is taken over every distance compatible with the topology of $X$. In the present work, we dropped the properness assumption, extending the above result for any continuous map $T$. We also apply our results to extend some previous formulas for the topological entropy of continuous endomorphisms of connected Lie groups proved in [4]. In particular, we prove that any linear transformation $T: V \to V$ over a finite dimensional vector space $V$ has null topological entropy.