紧子集与支撑测度的熵
Entropies of compact subsets and supported measures
- University of Science and Technology of China（中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究拓扑动力系统紧子集与支撑测度的各类拓扑熵的关系，证明三类熵的对应充要或蕴含关系，并给出反例说明其中一类关系的逆命题不成立。
AI中文摘要：
设$(X,T)$为拓扑动力系统，$(\boldsymbol{\textit{M}}(X),T_*)$为其诱导系统。对非空紧子集$K\boldsymbol{\textit{⊂}}X$，定义$\boldsymbol{\textit{M}}(K)$为支撑于$K$上的Borel概率测度集合。本文系统研究$(T,K)$与$(T_*,\boldsymbol{\textit{M}}(K))$的各类熵之间的关系，证明：$h_{\text{top}}^{\text{UC}}(T,K)>0$当且仅当$h_{\text{top}}^{\text{UC}}(T_*,\boldsymbol{\textit{M}}(K))=\boldsymbol{\textit{∞}}$；$h_{\text{top}}^{P}(T,K)>0$当且仅当$h_{\text{top}}^{P}(T_*,\boldsymbol{\textit{M}}(K))>0$；$h_{\text{top}}^{B}(T,K)>0$蕴含$h_{\text{top}}^{B}(T_*,\boldsymbol{\textit{M}}(K))>0$，其中$h_{\text{top}}^{\text{UC}}(T,K)$、$h_{\text{top}}^{P}(T,K)$、$h_{\text{top}}^{B}(T,K)$分别表示$K$的上容量拓扑熵、 packing拓扑熵、Bowen拓扑熵。此外，本文给出一个涉及非不变集的反例，表明第三个断言的逆命题一般不成立。
英文摘要：
Let $(X,T)$ be a topological dynamical system and $(\mathcal M(X),T_*)$ be its induced system. For a non-empty compact subset $K\subset X$, we define $\mathcal M(K)$ as the set of Borel probability measures supported on $K$. In this paper, we systematically study the relationship between various entropies of $(T,K)$ and of $(T_*,\mathcal M(K))$. We show that: \begin{equation*} \begin{aligned} & h_{\mathrm{top}}^{\mathrm{UC}}(T,K)>0 \iff h_{\mathrm{top}}^{\mathrm{UC}}(T_*,\mathcal{M}(K))=\infty, \qquad &h_{\mathrm{top}}^{P}(T,K)>0\iff h_{\mathrm{top}}^{P}(T_*,\mathcal{M}(K))>0, \qquad &h_{\mathrm{top}}^{B}(T,K)>0 \implies h_{\mathrm{top}}^{B}(T_*,\mathcal{M}(K))>0 , \end{aligned} \end{equation*} where $h_{\mathrm{top}}^{\mathrm{UC}}(T,K)$, $h_{\mathrm{top}}^{P}(T,K)$, and $h_{\mathrm{top}}^{B}(T,K)$ denote the upper capacity topological entropy, the packing topological entropy, and the Bowen topological entropy of $K$, respectively. Additionally, we present a counterexample involving a non-invariant set, demonstrating that the converse of the third assertion is not valid in general.
