Variational principles for topological pressures on subsets

TL;DR

This paper extends the concept of topological pressure for continuous transformations, establishing a variational principle and measure-theoretic framework that generalizes known results in topological dynamics.

Contribution

It introduces a measure-theoretic pressure for any invariant measure and proves a variational principle relating Bowen topological pressure to this measure-theoretic pressure.

Findings

Established a variational principle for topological pressure on subsets.

Defined measure-theoretic pressure for invariant measures.

Connected Bowen topological pressure with measure-theoretic pressure.

Abstract

The goal of this paper is to define and investigate those topological pressures, which is an extension of topological entropy presented by Feng and Huang [13], of continuous transformations. This study reveals the similarity between many known results of topological pressure. More precisely, the investigation of the variational principle is given and related propositions are also described. That is, this paper defines the measure theoretic pressure $P_{\mu}(T,f)$ for any $\mu\in{\mathcal M(X)}$, and shows that $P_B(T,f,K)=\sup\bigr\{P_{\mu}(T,f):{\mu}\in{\mathcalM(X)},{\mu}(K)=1\bigr\}$, where $K\subseteq X$ is a non-empty compact subset and $P_B(T,f,K)$ is the Bowen topological pressure on $K$. Furthermore, if $Z\subseteq X$ is an analytic subset, then $P_B(T,f,Z)=\sup\bigr\{P_B(T,f,K):K\subseteq Z\ \text{is compact}\bigr\}$. However, this analysis relies on more techniques of ergodic theory and topological dynamics.