饱和集、最优轨道和平衡态上的拓扑结构
Topological structures on saturated sets, optimal orbits and equilibrium states
摘要：
Pfister和Sullivan证明了：如果拓扑动力系统$(X,T)$满足几乎乘积性质和一致分离性质，则对于每个不变测度的非空紧%凸子集$K$，饱和集$G_{K}$的熵满足\begin{equation}\label{Bowen's topological entropy} h_{top}^{B}(T,G_{K})=\inf\{h(T,μ):μ\in K\}, \end{equation}，其中$h_{top}^{B}(T,G_{K})$是$T$在$G_{K}$上的Bowen拓扑熵，$h(T,μ)$是$μ$的Kolmogorov-Sinai熵.本文用上容量熵和填充熵代替Bowen拓扑熵，研究了$G_{K}$的拓扑复杂性，得到了如下公式：\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*}其中$h_{top}^{UC}(T,G_{K})$是$G_{K}$上的$T$的上容量熵，$h_{top}^{P}(T,G_{K})$是$G_{K}.$上的$T$的填充熵。
英文摘要：
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,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.
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