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The conditional variational principle for maps with the pseudo-orbit tracing property

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  • 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

    $\mathscr{E}_n(x) = \frac{1}{n}\sum\limits_{i = 0}^{n-1}δ_{f^ix},$

    where $δ_y$ 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

    $\Delta_{sub}(I): = \left\{ {x \in X:V(x)\subset I} \right\},$


    $\Delta_{cap}(I): = \left\{ {x \in X:V(x)\cap I≠\emptyset } \right\}.$

    in a dynamical system with the pseudo-orbit tracing property, where $I$ is a certain subset of $\mathscr M_{\rm inv}(X,f)$.

    Mathematics Subject Classification: Primary: 37B40; Secondary: 37C45.


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