{
  "abstract": "Let $T_n(x)=nx\\pmod 1$ on $\\T=\\mathbb R/\\mathbb Z$, and let $Σ\\subset\\mathbb N$ be a finitely generated nonlacunary multiplicative semigroup. We prove that, for every $1\\le s\\le\\infty$, there is an open dense set of potentials $f\\in W^{1,s}(\\T)$ such that every measure that maximizes or minimizes $\\int f\\,dμ$ over the $Σ$-invariant probability measures satisfies $h_μ(T_r)=0$ for all $r\\inΣ\\setminus\\{1\\}$. For each fixed $s$, a single dense $G_δ$ subset of $W^{1,s}(\\T)$ works simultaneously for all finitely generated nonlacunary multiplicative semigroups. The proof combines Rudolph--Johnson entropy rigidity with periodic-grid perturbations that generically exclude Haar measure from the optimizing faces. In particular, without any uniqueness assumption, the result applies to simultaneous $T_p,T_q$-invariance whenever $p,q\\ge2$ are multiplicatively independent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.24503",
  "author_total": 1,
  "authors": [
    {
      "name": "Hang Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b7e40ara9f51f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.24503",
  "fulltext_url": null,
  "id": "b7e40ara9f51f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.24503",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7204275969",
  "pdf_url": "https://arxiv.org/pdf/2608.24503",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b7e40ara9f51f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.24503",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.24503"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Generic Zero-Entropy Optimization for Finitely Generated Nonlacunary Actions on the Circle",
  "title_text": "Generic Zero-Entropy Optimization for Finitely Generated Nonlacunary Actions on the Circle",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/b7e40ara9f51f",
  "venue": "arXiv",
  "versions": []
}