A variational principle of topological pressure on subsets for amenable group actions

Abstract

<p style='text-indent:20px;'>In this paper, we establish a variational principle for topological pressure on compact subsets in the context of amenable group actions. To be precise, for a countable amenable group action on a compact metric space, say <inline-formula><tex-math id="M1">\begin{document}$ G\curvearrowright X $\end{document}</tex-math></inline-formula>, for any potential <inline-formula><tex-math id="M2">\begin{document}$ f\in C(X) $\end{document}</tex-math></inline-formula>, we define and study topological pressure on an arbitrary subset and measure theoretic pressure for any Borel probability measure on <inline-formula><tex-math id="M3">\begin{document}$ X $\end{document}</tex-math></inline-formula> (not necessarily invariant); moreover, we prove a variational principle for this topological pressure on a given nonempty compact subset <inline-formula><tex-math id="M4">\begin{document}$ K\subseteq X $\end{document}</tex-math></inline-formula>.