Variational principle for weighted amenable topological pressure

TL;DR

This paper develops a variational principle for weighted amenable topological pressure in the context of group actions, linking it to measure-theoretic entropy and equilibrium states.

Contribution

It introduces the concept of weighted amenable topological pressure for factor maps and establishes a variational principle relating it to entropy and equilibrium states.

Findings

Established a variational principle for weighted amenable topological pressure.

Connected weighted topological pressure with measure-theoretic entropy.

Analyzed equilibrium states of weighted topological pressure.

Abstract

This paper aims to investigate the thermodynamic formalism of weighted amenable topological pressure for factor maps of amenable group actions. Following the approach of Tsukamoto [\emph{Ergodic Theory Dynam. Syst.} \textbf{43}(2023), 1004-1034.], we introduce the notion of weighted amenable topological pressure for factor maps of amenable group actions, and establish a variational principle for it. As the application of variational principle, we show weighted amenable measure-theoretic entropy can be determined by weighted amenable topological pressure. Equilibrium states of weighted topological pressure are also involved.