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PreprintarXiv licencearXiv

Return Time Sets of Minimal Points for Countable Group Actions

  • Jian Li
  • Yini Yang
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Abstract

For a countably infinite discrete group G, we study subsets of G that contain a neighbourhood return-time set of a minimal point in a G-flow X. For e∈ F⊆ G, we characterize this property by the existence of e∈ A⊆ F such that IntK(A):={t∈ G:Kt⊆ A} is right syndetic for every finite K⊆ A, and equivalently by the support of a minimal binary configuration in the Bernoulli shift 2G. Additional conditions on IntK(A) IntK(A)−1 characterize realization by minimal flows that are weakly mixing of all finite orders or strongly mixing: for every such K, the former requires that for each finite H⊆ G there is c∈ G with HcH⊆ IntK(A) IntK(A) −1, whereas the latter requires that IntK(A)IntK(A)−1 be cofinite. We also construct, for every compact metrizable minimal G-flow, a free minimal, strongly mixing Cantor flow disjoint from it.

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Cite

BibTeXRIS
BibTeX
@misc{li2026return,
  title = {{Return Time Sets of Minimal Points for Countable Group Actions}},
  author = {Jian Li and Yini Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30554},
  eprint = {2610.30554},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30554},
}
RIS
TY  - UNPB
TI  - Return Time Sets of Minimal Points for Countable Group Actions
AU  - Jian Li
AU  - Yini Yang
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30554
UR  - https://arxiv.org/abs/2610.30554
AB  - For a countably infinite discrete group G, we study subsets of G that contain a neighbourhood return-time set of a minimal point in a G-flow X. For e∈ F⊆ G, we characterize this property by the existence of e∈ A⊆ F such that Int_K(A):={t∈ G:Kt⊆ A} is right syndetic for every finite K⊆ A, and equivalently by the support of a minimal binary configuration in the Bernoulli shift 2^G. Additional conditions on Int_K(A) Int_K(A)^−1 characterize realization by minimal flows that are weakly mixing of all finite orders or strongly mixing: for every such K, the former requires that for each finite H⊆ G there is c∈ G with HcH⊆ Int_K(A) Int_K(A) ^−1, whereas the latter requires that Int_K(A)Int_K(A)^−1 be cofinite. We also construct, for every compact metrizable minimal G-flow, a free minimal, strongly mixing Cantor flow disjoint from it.
ER  -