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The first author is partially supported by Project MTM2017-89686-P (AEI/FEDER, UE); the second author is partially supported by a Canon Foundation in Europe Research Fellowship; the third author is supported by the FWF Project P31950-N35; the forth author is partially supported by JSPS KAKENHI Grant number 17K14195 and 20K03620.

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Wild Cantor actions

Publicado en:Journal Of The Mathematical Society Of Japan. 74 (2): 447-472 - 2022-04-01 74(2), DOI: 10.2969/jmsj/85748574

Autores: Alvarez Lopez, Jesus; Lio, Ramon Barral; Lukina, Olga; Nozawa, Hiraku

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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.

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