Topological structures on saturated sets, optimal orbits and equilibrium states

In this paper, we investigate topological complexity of saturated set from the viewpoints upper capacity entropy and packing entropy. We obtain that if a dynamical system $ (X, T) satisfies almost product property, then for each nonempty compact convex subset K invariant measures, following two formulas hold G_K $:$ \begin{equation*} h_{top}^{UC}(T, G_{K}) = h_{top}(T, X)\ \mathrm{and}\ h_{top}^{P}(T, \sup\{h(T, \mu):\mu\in K\}, \end{equation*} $where is T on G_{K}, h(T, \mu) Kolmogorov-Sinai \mu $. As applications, transitive Anosov diffeomorphism manifold, use above results to quantify optimal orbits equilibrium states both real matrix valued potentials.