Jesús ÁLVAREZ LÓPEZ, Ramon BARRAL LIJO, Olga LUKINA, Hiraku NOZAWA

J. Math. Soc. Japan Advance Publication, 1-26, (July, 2021) DOI: 10.2969/jmsj/85748574
KEYWORDS: group actions, Cantor sets, equicontinuous actions, group actions on rooted trees, wreath products, profinite groups, stabilizer direct limit group, centralizer direct limit group, the alternating group, the cyclic group, 37E25, 20E08, 20E15, 20E18, 20E22, 22F05, 20F22, 22F50, 37B05, 57R30, 57R50

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.