Topological structures on saturated sets, optimal orbits and equilibrium states

TL;DR

This paper explores the topological complexity of saturated sets in dynamical systems by analyzing various entropy measures, extending previous results and removing the need for the uniform separation property.

Contribution

It generalizes the understanding of entropy on saturated sets by replacing Bowen's entropy with upper capacity and packing entropy, without requiring the uniform separation property.

Findings

Upper capacity entropy of saturated sets equals the topological entropy of the system.

Packing entropy of saturated sets equals the supremum of measure-theoretic entropies.

Results hold without the uniform separation property.

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,\mu):\mu\in K\}, \end{equation} where $h_{top}^{B}(T,G_{K})$ is Bowen's topological entropy of $T$ on $G_{K}$, and $h(T,\mu)$ is the Kolmogorov-Sinai entropy of $\mu$. 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,\mu):\mu\in K\}, \end{equation*} where $h_{top}^{UC}(T,G_{K})$ is the upper capacity entropy of $T$ on $G_{K}$ and $h_{top}^{P}(T,G_{K})$ is the packing entropy of $T$ on $G_{K}.$ In the proof of these two formulas, uniform separation property is unnecessary.