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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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Lio, Ramon BarralAuthor

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Wild Cantor actions

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

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

Affiliations

Ritsumeikan Univ, Dept Math Sci, Coll Sci & Engn, 1 Chome 1 1 Nojihigashi, Kusatsu, Shiga 5258577, Japan - Author
Ritsumeikan Univ, Res Org Sci & Technol, 1 Chome 1 1 Nojihigashi, Kusatsu, Shiga 5258577, Japan - Author
Univ Santiago de Compostela, Dept Math, Fac Math, Santiago De Compostela 15782, Spain - Author
Univ Vienna, Fac Math, Oskar Morgenstern Pl 1, A-1090 Vienna, Austria - Author

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.

Keywords

Cantor setsCentralizer direct limit groupEquicontinuous actionsGroup actionsGroup actions on rooted treesProfinite groupsStabilizer direct limit groupThe alternating groupThe cyclic grouWreath products

Quality index

Bibliometric impact. Analysis of the contribution and dissemination channel

The work has been published in the journal Journal Of The Mathematical Society Of Japan due to its progression and the good impact it has achieved in recent years, according to the agency Scopus (SJR), it has become a reference in its field. In the year of publication of the work, 2022, it was in position , thus managing to position itself as a Q1 (Primer Cuartil), in the category Mathematics (Miscellaneous).

From a relative perspective, and based on the normalized impact indicator calculated from the Field Citation Ratio (FCR) of the Dimensions source, it yields a value of: 1.94, which indicates that, compared to works in the same discipline and in the same year of publication, it ranks as a work cited above average. (source consulted: Dimensions Jun 2025)

Specifically, and according to different indexing agencies, this work has accumulated citations as of 2025-06-10, the following number of citations:

  • WoS: 1
  • Scopus: 3
  • OpenCitations: 1

Impact and social visibility

From the perspective of influence or social adoption, and based on metrics associated with mentions and interactions provided by agencies specializing in calculating the so-called "Alternative or Social Metrics," we can highlight as of 2025-06-10:

  • The use of this contribution in bookmarks, code forks, additions to favorite lists for recurrent reading, as well as general views, indicates that someone is using the publication as a basis for their current work. This may be a notable indicator of future more formal and academic citations. This claim is supported by the result of the "Capture" indicator, which yields a total of: 1 (PlumX).

Leadership analysis of institutional authors

This work has been carried out with international collaboration, specifically with researchers from: Austria; Japan.