IC-rigidity for abelian actions — Draconitic Research

Abstract

Let G be a group acting continuously on a compact metric space X. This naturally induces an action of G on the space 𝒦(X) of nonempty compact subsets of X. In [KS26], Kra and Schmieding define a system (X,G) to be IC-rigid if the union of the supports of the G-invariant Borel probability measures on 𝒦(X) is as small as possible. They leave open the question of existence of IC-rigid abelian systems. We show the existence of IC-rigid actions for three classes of abelian groups: mixing rank-one ℤ-actions, horocycle flows, and certain mixing actions on a Cantor space of G=⨁_n=1^∞G_n, where (G_n)_n is any sequence of nontrivial finite abelian groups. In the course of proving these results, we introduce the notion of weakly asymptotic pairs and establish its close connection to topological minimal self-joinings.

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BibTeX RIS

BibTeX

@misc{kanigowski2026icrigidity,
  title = {{IC-rigidity for abelian actions}},
  author = {Adam Kanigowski and Kosma Kasprzak and Miriam Lorenzo-Laguno},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36397},
  eprint = {2609.36397},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36397},
}

RIS

TY  - UNPB
TI  - IC-rigidity for abelian actions
AU  - Adam Kanigowski
AU  - Kosma Kasprzak
AU  - Miriam Lorenzo-Laguno
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.36397
UR  - https://arxiv.org/abs/2609.36397
AB  - Let G be a group acting continuously on a compact metric space X. This naturally induces an action of G on the space 𝒦(X) of nonempty compact subsets of X. In [KS26], Kra and Schmieding define a system (X,G) to be IC-rigid if the union of the supports of the G-invariant Borel probability measures on 𝒦(X) is as small as possible. They leave open the question of existence of IC-rigid abelian systems. We show the existence of IC-rigid actions for three classes of abelian groups: mixing rank-one ℤ-actions, horocycle flows, and certain mixing actions on a Cantor space of G=⨁_n=1^∞G_n, where (G_n)_n is any sequence of nontrivial finite abelian groups. In the course of proving these results, we introduce the notion of weakly asymptotic pairs and establish its close connection to topological minimal self-joinings.
ER  -

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