Topological entropy in totally disconnected locally compact groups
Topological entropy in totally disconnected locally compact groups
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完全不连通的局部紧群中的拓扑熵
DOI：
10.1017/etds.2015.139
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发表时间：
2015
影响因子：
0.9
通讯作者：
Simone Virili
中科院分区：
文献类型：
--
作者：
A. Giordano Bruno;Simone Virili
设$G$为拓扑群，$\unicode[STIX]{x1D719}$为$G$的连续自同态，$H$为$G$的闭$\unicode[STIX]{x1D719}$不变子群。我们研究拓扑熵是否为加性不变量，即$$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$，其中$\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$为$\unicode[STIX]{x1D719}$诱导的映射。我们集中研究$G$完全断开的局部紧致的情况，而$H$是紧致的或正常的。在这些假设下，我们证明了上述可加性在$\unicode[STIX]{x1D719}H=H$和$\ker (\unicode[STIX]{x1D719})\leq H$时成立。作为应用，我们给出了尺度$s(\unicode[STIX]{x1D719})$的动态解释，表明$\log s(\unicode[STIX]{x1D719})$是由$\unicode[STIX]{x1D719}$引起的合适映射的拓扑熵。最后给出了等式$\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$成立的充分必要条件。
Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$ -invariant subgroup of $G$ . We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$ . We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$ . As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$ . Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.