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

TL;DR

This paper establishes conditional variational principles for topological entropy in dynamical systems with the pseudo-orbit tracing property, linking entropy to sets of points with specific limit measure behaviors.

Contribution

It introduces new variational principles for entropy related to sets of points characterized by their limit measures in systems with pseudo-orbit tracing.

Findings

Derived variational principles for entropy of measure-based point sets

Connected topological entropy to limit measure sets in pseudo-orbit systems

Extended entropy analysis to subsets defined by measure intersection properties

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 $\mathscr M_{\rm inv}(X,f)$.