| Abstrakt: |
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. [ABSTRACT FROM AUTHOR] |
