2026/07/22 by Chunlin Liu
#math.DS
Let G be a countably infinite discrete group and let π:(X,μ,G)→(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 (si)i≥ 1 in G, there is a Cantor set K(si)⊆ X whose distinct points x,x' satisfy \liminfi→∞ρ(si x,si x')=0, \limsupi→∞ρ(si x,si 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.