paper

Topological structures on saturated sets, optimal orbits and equilibrium states

arXiv:2012.09482

Abstract

Pfister and Sullivan proved that if a topological dynamical system $(X, T)$ satisfies almost product property and uniform separation property, then for each nonempty compact %convex subset $K$ of invariant measures, the entropy of saturated set $G_{K}$ satisfies \begin{equation}\label{Bowen's topological entropy} h_{top}^{B}(T,G_{K})=\inf\{h(T,Î¼):Î¼\in K\}, \end{equation} where $h_{t o p}^{B} (T, G_{K})$ is Bowen's topological entropy of $T$ on $G_{K}$ , and $h (T, I^¼)$ is the Kolmogorov-Sinai entropy of $I^¼$ . In this paper, we investigate topological complexity of $G_{K}$ by replacing Bowen's topological entropy with upper capacity entropy and packing entropy and obtain the following formulas: \begin{equation*} h_{top}^{UC}(T,G_{K})=h_{top}(T,X)\ \mathrm{and}\ h_{top}^{P}(T,G_{K})=\sup\{h(T,Î¼):Î¼\in K\}, \end{equation*} where $h_{t o p}^{U C} (T, G_{K})$ is the upper capacity entropy of $T$ on $G_{K}$ and $h_{t o p}^{P} (T, G_{K})$ is the packing entropy of $T$ on $G_{K} .$ In the proof of these two formulas, uniform separation property is unnecessary.