Variational principle for conditional pressure with subadditive potential (Q2891179)

Consider two compact metric dynamical systems \((X,T)\) and \((Y,S)\) and denote by \(\pi\colon X\to Y\) the factor map. Let \(d_n(x,y)=\max\{d(T^i(x),T^i(y))\colon 0\leq i\leq n-1\}\) be a new metric on \(X\). A set \(E\subset K\) is said to be \((n,\epsilon)\)-separated if \(d_n(x,y)>\epsilon\) for any \(x,y\in E\). A sequence \(\mathcal F=\{f_n\colon n\in\mathbb N\}\subset C(X)\) is called a subadditive potential if NEWLINE\[NEWLINEf_{n+m}(x)\leq f_n(x)+f_m(T^m(x)).NEWLINE\]NEWLINE Denote by \(\mathcal M(X,T)\) the set of invariant probability measures on \((X,T)\). For \(\mu\in\mathcal M(X,T)\) and a subadditive potential \(\mathcal F=\{f_n\}\), NEWLINE\[NEWLINE\mathcal F_*(\mu)=\lim_{n\to\infty}{1\over n}\int f_n\,d\mu.NEWLINE\]NEWLINE The metrical conditional entropy is defined as follows: Let \(\mathcal A\) be the \(\sigma\)-algebra generated by \(\{\pi^{-1}(y)\colon y\in Y\}\) and let \(\xi\) be a partition of \(X\). Then define NEWLINE\[NEWLINEh_\mu(T,\xi\mid\mathcal A) =\lim_{n\to\infty}H_\mu\Bigl(\bigvee_{i=0}^{n-1}T^{-i}\xi\mid\mathcal A\Bigr),NEWLINE\]NEWLINE where \(H_\mu(\xi\mid\mathcal A)\) is the usual conditional metrical entropy. Then the metrical conditional entropy is defined by NEWLINE\[NEWLINEh_\mu(T,X\mid Y)=\sup \{h_\mu(T,\xi\mid\mathcal A)\colon \text{\(\xi\) is a finite partition of \(X\)}\}.NEWLINE\]NEWLINE For \(y\in E\), consider NEWLINE\[NEWLINEP_n(T,\mathcal F,\epsilon,y)=\sup_E\left\{\sum_{x\in E}e^{f_n(x)} \colon E\text{ is an }(n,\epsilon)\text{-separated subset of } \pi^{-1}(y)\right\}.NEWLINE\]NEWLINE Then NEWLINE\[NEWLINE\begin{aligned} P(T,\mathcal F,\epsilon,y)=&\limsup_{n\to\infty}{1\over n}P_n(T,\mathcal F,\epsilon,y),\\ P(T,\mathcal F,\epsilon, Y)=&\sup_{y\in Y}P(T,\mathcal F,\epsilon,y). \end{aligned}NEWLINE\]NEWLINE Finally, the topological conditional pressure is defined by NEWLINE\[NEWLINEP(T,\mathcal F,Y)=\lim_{\epsilon\to0}P(T,\mathcal F,\epsilon,Y).NEWLINE\]NEWLINE This pressure has properties similar to the usual topological pressure which is defined to be \(f_n(x)=\sum_{i=0}^{n-1}f(T^i(x))\). The main result of this article goes as follows:NEWLINENEWLINE Let \(\pi\colon(X,T)\to(Y,S)\) be a factor map between two compact metric dynamical systems and \(\mathcal F=\{f_n\}_{n=1}^\infty\) be a subadditive potential on \(X\). Then NEWLINE\[NEWLINEP(T,\mathcal F,Y)= \begin{cases} -\infty &\text{if }\mathcal F_*(\mu)=-\infty \text{ for all } \mu\in\mathcal M(X,T),\\ \displaystyle\sup_{\mu\in\mathcal M(X,T)\atop F_*(\mu)\neq-\infty}\left\{ h_\mu(T,X\mid Y)+\mathcal F_*(\mu)\right\}&\text{otherwise.} \end{cases}NEWLINE\]