发表机构
Dalian University of Technology; Institute of Mathematics, Polish Academy of Sciences（大连理工大学; 波兰科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究可数无限离散群\(G\)的非平凡相对混合扩张，证明存在常数\(\delta>0\)，对\(G\)中每个单射序列\((s_i)\)，能找到满足特定条件的康托集\(K_{(s_i)}\)，还给出了该结论对索菲克群及相关群作用的应用。
AI 中文摘要
设\(G\)为可数无限离散群，\(\pi:(X,\mu,G)\to(Y,\nu,G)\)为非平凡相对混合扩张，其中\(X\)是紧致可度量\(G\)-空间。我们证明存在常数\(\delta>0\)，使得对于\(G\)中每个单射序列\((s_i)_{i\geq1}\)，存在康托集\(K_{(s_i)}\subseteq X\)，其不同点\(x,x'\)满足\(\liminf_{i\to\infty}\rho(s_ix,s_ix') = 0\)，\(\limsup_{i\to\infty}\rho(s_ix,s_ix')>\delta\)。该方法还产生高阶混沌康托集。作为主要应用，对于索菲克群\(G\)，正拓扑索菲克熵意味着上述结论，回答了黄、李和叶的一个问题。对于具有正罗赫林熵的本质自由不变测度的任意可数无限离散群的作用，同样结论也成立。
英文摘要
Let $G$ be a countably infinite discrete group and let $ π:(X,μ,G)\to(Y,ν,G) $ be a nontrivial relatively mixing extension, where $X$ is a compact metrizable $G$-space. We prove that there exists a constant $δ>0$ such that, for every injective sequence $(s_i)_{i\geq 1}$ in $G$, there is a Cantor set $K_{(s_i)}\subseteq X$ whose distinct points $x,x'$ satisfy \[ \liminf_{i\to\infty}ρ(s_i x,s_i x')=0, \qquad \limsup_{i\to\infty}ρ(s_i x,s_i x')>δ. \] The method also yields higher-order scrambled Cantor sets. As a principal application, for a sofic group $G$, positive topological sofic entropy implies the preceding conclusion, answering a question of Huang, Li, and Ye. The same conclusion also holds for actions of arbitrary countably infinite discrete groups admitting an essentially free invariant measure of positive Rokhlin entropy. In contrast, we construct a hereditary subshift over a residually finite group with positive naive topological entropy for which all points converge to a common fixed point along a prescribed injective sequence. This answers a question of García-Ramos and Li negatively even for sofic groups.
CommentsThe revised version includes a counterexample showing that positive naive topological entropy is insufficient to guarantee Li--Yorke chaos along every prescribed infinite sequence. We welcome any comments, suggestions, or discussion regarding our manuscript
