Topological structures on saturated sets, optimal orbits and equilibrium states

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)\ $\mathrm{a}\mathrm{n}\mathrm{d}$
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 $G_K\mathrm{.}$ In the proof of these two formulas, uniform separation property is unnecessary.