{
  "abstract": "This paper studies disjointness and \\(\\mathscr{F}\\)-product recurrence with respect to zero-entropy systems. First, we prove that, for a point whose orbit closure has zero topological entropy, \\[ \\text{distality}\\rightleftharpoons \\Finf\\PRzero\\rightleftharpoons \\Fpubd\\PRzero \\rightleftharpoons \\Fps\\PRzero \\rightovernotleft \\Fs\\PRzero. \\] 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^{\\perp}\\subsetneq \\Mzero^{\\perp}\\), 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 \\(\\Fps\\PRzero\\) point which is not \\(\\Fpubd\\PRzero\\), i.e., $\\Fps\\PRzero$ $\\nRightarrow \\Fpubd\\PRzero$, 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].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25094",
  "author_total": 1,
  "authors": [
    {
      "name": "Xinxing Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/20mz10ksnkwhr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25094",
  "fulltext_url": null,
  "id": "20mz10ksnkwhr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25094",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214104248",
  "pdf_url": "https://arxiv.org/pdf/2609.25094",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/20mz10ksnkwhr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25094",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25094"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "On disjointness and $\\mathscr{F}$-product recurrence with respect to zero entropy systems",
  "title_text": "On disjointness and ℱ-product recurrence with respect to zero entropy systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/20mz10ksnkwhr",
  "venue": "arXiv",
  "versions": []
}