Wild Cantor actions

TL;DR

This paper classifies minimal equicontinuous actions on Cantor sets using discriminant, stabilizer, and centralizer groups, and constructs new examples via actions on rooted trees to illustrate the classification.

Contribution

It introduces new families of minimal equicontinuous actions on Cantor sets constructed from groups acting on rooted trees, expanding the understanding of their classification.

Findings

New examples of minimal equicontinuous actions on Cantor sets.

Illustration of classification via stabilizer and centralizer groups.

Applications to dynamical systems and foliation minimal sets.

Abstract

The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations.