Wild Cantor actions

DOI10.2969/JMSJ/85748574arXiv2010.00498MaRDI QIDQ6350324

Jesus A. Álvarez López, Olga Lukina, Hiraku Nozawa, Ramón Barral Lijó

Publication date: 1 October 2020

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.

Mathematics Subject Classification ID

Other classes of groups defined by subgroup chains (20F22) Chains and lattices of subgroups, subnormal subgroups (20E15) Extensions, wreath products, and other compositions of groups (20E22) Foliations in differential topology; geometric theory (57R30) Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) (37B05) Dynamical systems involving maps of trees and graphs (37E25) Groups acting on trees (20E08) Limits, profinite groups (20E18) Differential topological aspects of diffeomorphisms (57R50) General theory of group and pseudogroup actions (22F05) Groups as automorphisms of other structures (22F50)

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