Topological structures on saturated sets, optimal orbits and equilibrium states

Abstract

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