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{Bowens topological entropy} h_{top}^{B}(T,G_{K})=inf{h(T,mu):muin K}, end{equation} where $h_{top}^{B}(T,G_{K})$ is Bowens 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 Bowens 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):muin 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.