The Conditional Variational Principle for Maps with the Pseudo-orbit Tracing Property

Abstract

Let $(X,d,f)$ be a topological dynamical system, where $(X,d)$ is a compact metric space and $f:X\to X$ is a continuous map. We define $n$-ordered empirical measure of $x\in X$ by \begin{align*} \mathscr{E}_n(x)=\frac{1}{n}\sum\limits_{i=0}^{n-1}\delta_{f^ix}, \end{align*} where $\delta_y$ is the Dirac mass at $y$. Denote by $V(x)$ the set of limit measures of the sequence of measures $\mathscr{E}_n(x).$ In this paper, we obtain conditional variational principles for the topological entropy of \begin{align*} \Delta_{sub}(I)=\left\{x\in X:V(x)\subset I\right\}, \end{align*} and \begin{align*} \Delta_{cap}(I)=\left\{x\in X:V(x)\cap I\neq\emptyset \right\}. \end{align*} in a transitive dynamical system with the pseudo-orbit tracing property, where $I$ is a certain subset of $\mathscr M_{\rm inv}(X,f)$.