Variational inequality for conditional pressure on a Borel subset (Q433538)

Topological entropy and pressure have played very important roles in the study of modern dynamical systems. Let \(T:X\rightarrow X\) be a continuous map on a compact metric space \(X\). Given a continuous function \(\varphi\) on \(X\) and a closed \(T\)-invariant subset \(G\) of \(X\), the authors prove the following inequalities NEWLINE\[NEWLINEP_{G}(T, \varphi)\leq \sup_{\mu\in \mathcal{M}_T} \{h_{\mu}(T\mid <G>)+\int \varphi d\mu \}\leq \max \{P_{G}(T, \varphi),P_{\overline{X\setminus G}}(T, \varphi)\}NEWLINE\]NEWLINE where \( \mathcal{M}_T\) is the space of all \(T\)-invariant measures, \(P_{A}(T, \varphi)\) denotes the topological pressure of \(T\) w.r.t. \(\varphi\) on a set \(A\), and \(h_{\mu}(T\mid <G>)\) is the conditional entropy of \(T\) w.r.t. \(\mu\) and the partition \(<G>:=\{G, x\setminus G\}\). This result extends the third author's result [Taiwanese J. Math. 12, No. 7, 1791--1803 (2008; Zbl 1221.37027)] to pressure.