{
  "abstract": "We show that beta-shifts whose lexicographically largest point is a Sturmian sequence do not have typical periodic optimization (TPO): on each such beta-shift there is a non-empty open set of Lipschitz functions whose unique maximizing measure is the Sturmian measure, which is not supported on a periodic orbit. These are the first known examples of beta-shifts without TPO, and, since they have the specification property, the first known examples of shift spaces with specification but without TPO. The corresponding beta-transformations do not have TPO in any space of Hölder functions on the interval. The main ingredient in the proofs is a rigidity property of Sturmian subshifts: modulo constants and Lipschitz coboundaries, the space of Lipschitz functions on such a subshift is one-dimensional. We also show that the set of $β>1$ for which the beta-shift or the beta-transformation does not have TPO is dense in $(1,\\infty)$, but has zero packing dimension. The density is proved by a tuning construction, analogous to Douady--Hubbard tuning in the Mandelbrot set, which produces non-Sturmian subshifts with the same rigidity property. Finally, we show that every beta-shift has typical zero-entropy optimization: for an open dense set of Lipschitz functions, the maximizing measure is unique and its support has zero topological entropy.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03619",
  "author_total": 4,
  "authors": [
    {
      "name": "Wen Huang"
    },
    {
      "name": "Oliver Jenkinson"
    },
    {
      "name": "Leiye Xu"
    },
    {
      "name": "Yiwei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0h18kkwpe9xs7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03619",
  "fulltext_url": null,
  "id": "0h18kkwpe9xs7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03619",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7220390819",
  "pdf_url": "https://arxiv.org/pdf/2610.03619",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0h18kkwpe9xs7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.03619",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03619"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Beta-shifts without typical periodic optimization",
  "title_text": "Beta-shifts without typical periodic optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/0h18kkwpe9xs7",
  "venue": "arXiv",
  "versions": []
}