论文标题
饱和集，最佳轨道和平衡状态的拓扑结构
Topological structures on saturated sets, optimal orbits and equilibrium states
论文作者
论文摘要
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，t，g_ {k}）= \ inf \ {h（t，μ）：μ\ in k \}，\ end {equation}，其中$ h_ {pop}^{pop}^{b}（t，g_ {k}） $ h（t，μ）$是$μ$的kolmogorov-sinai熵。在本文中，我们通过用高容量熵和包装熵代替鲍恩的拓扑熵来研究$ g_ {k} $的拓扑复杂性，并获得以下公式：\ begin {equination {equation*} h_ {top}^{uc}（t，t，g_ {k}）= h_ {top}（t，t，x）\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ h_ {top}^{pop}^{p}（t，g_ {k {k {k {k {k}）= \ sup \ sup \ sup \ \ {h（t，t，μ） $ h_ {top}^{uc}（t，t，g_ {k}）$是$ g_ {k {k} $和$ h_ {top}^{p}^{p}（t，g_ {k}）$的$ t $的上能熵，是$ g_ {k_ {k_ {
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,μ):μ\in K\}, \end{equation} where $h_{top}^{B}(T,G_{K})$ is Bowen's topological entropy of $T$ on $G_{K}$, and $h(T,μ)$ is the Kolmogorov-Sinai entropy of $μ$. 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{and}\ h_{top}^{P}(T,G_{K})=\sup\{h(T,μ):μ\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 $G_{K}.$ In the proof of these two formulas, uniform separation property is unnecessary.
