On disjointness and ℱ-product recurrence with respect to zero entropy systems

Abstract

This paper studies disjointness and ℱ-product recurrence with respect to zero-entropy systems. First, we prove that, for a point whose orbit closure has zero topological entropy, distalityrightleftharpoons FinfPRzerorightleftharpoons FpubdPRzero rightleftharpoons FpsPRzero rightovernotleft FsPRzero. We also construct a minimal topological dynamical system with uniform positive entropy which is disjoint from all zero-entropy M-systems but is not disjoint from some zero-entropy E-system, i.e., Ezero^⊥subsetneq Mzero^⊥, giving an affirmative answer to a question of \cite[W.~Huang, K.~K. Park and X.~Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France \textbf{135} (2007), 259--282]. Moreover, the same construction shows that there exists an FpsPRzero point which is not FpubdPRzero, i.e., FpsPRzero nRightarrow FpubdPRzero, giving a negative answer to a question of \cite[P.~Oprocha and G.~H. Zhang, On weak product recurrence and synchronization of return times, Adv. Math. \textbf{244} (2013), 395--412].

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BibTeX RIS

BibTeX

@misc{wu2026disjointness,
  title = {{On disjointness and $\mathscrF$-product recurrence with respect to zero entropy systems}},
  author = {Xinxing Wu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25094},
  eprint = {2609.25094},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25094},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On disjointness and ℱ-product recurrence with respect to zero entropy systems
AU  - Xinxing Wu
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.25094
UR  - https://arxiv.org/abs/2609.25094
AB  - This paper studies disjointness and ℱ-product recurrence with respect to zero-entropy systems. First, we prove that, for a point whose orbit closure has zero topological entropy, distalityrightleftharpoons FinfPRzerorightleftharpoons FpubdPRzero rightleftharpoons FpsPRzero rightovernotleft FsPRzero. We also construct a minimal topological dynamical system with uniform positive entropy which is disjoint from all zero-entropy M-systems but is not disjoint from some zero-entropy E-system, i.e., Ezero^⊥subsetneq Mzero^⊥, giving an affirmative answer to a question of \cite[W.~Huang, K.~K. Park and X.~Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France \textbf{135} (2007), 259--282]. Moreover, the same construction shows that there exists an FpsPRzero point which is not FpubdPRzero, i.e., FpsPRzero nRightarrow FpubdPRzero, giving a negative answer to a question of \cite[P.~Oprocha and G.~H. Zhang, On weak product recurrence and synchronization of return times, Adv. Math. \textbf{244} (2013), 395--412].
ER  -

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