Variational principles for relative local pressure with subadditive potentials

Abstract

We prove two relative local variational principles of topological pressure functions \documentclass[12pt]{minimal}\begin{document}$P(T,\mathcal {F},\mathcal {U},y)$\end{document}P(T,F,U,y) and \documentclass[12pt]{minimal}\begin{document}$P(T,\mathcal {F},\mathcal {U}|Y)$\end{document}P(T,F,U|Y) for a given factor map π: (X, T) → (Y, S) between two topological dynamical systems, an open cover \documentclass[12pt]{minimal}\begin{document}$\mathcal {U}$\end{document}U of X and a subadditive potential \documentclass[12pt]{minimal}\begin{document}$\mathcal {F}$\end{document}F in \documentclass[12pt]{minimal}\begin{document}$C(X,\mathbb {R})$\end{document}C(X,R). By proving the upper semi-continuity and affinity of the entropy maps \documentclass[12pt]{minimal}\begin{document}$\mu \rightarrow h_{\mu }(T,\mathcal {U}\mid Y)$\end{document}μ→hμ(T,U∣Y) and \documentclass[12pt]{minimal}\begin{document}$\mu \rightarrow h^+_{\mu }(T,\mathcal {U}\mid Y)$\end{document}μ→hμ+(T,U∣Y) on the space of all invariant Borel probability measures, we show that the relative local topological pressure with subadditive potentials determines the local measure-theoretic conditional entropies.