{
  "abstract": "Let $T$ be the Gauss map. For each $α\\in [0,1]$, let $E(α)$ be the survivor set consisting of all $x\\in [0,1]$ whose orbit $\\{T^k(x):k\\ge 0\\}$ never enters the interval $(α,1]$. We prove that the set-valued function $α\\mapsto E(α)$ is increasing and locally constant almost everywhere. Moreover, we explicitly characterize all intervals where the function $α\\mapsto E(α)$ remains constant. We also describe the points at which the map $α\\mapsto E(α)$ is not locally constant. Further, we obtain the Hausdorff dimension of $E(α)$ for all $α\\in [0,1]$. We prove that the function $α\\mapsto \\dim_H(E(α))$ forms a devil's staircase on $(0,1]$, but is not continuous at $0$. This discontinuity implies that the result of Raith [Acta Math. Univ. Comenian. 63 (1994), 39-53] fails for piecewise monotone interval maps with infinitely many branches. Finally, we give an application quantifying the set of points sharing identical $α$-continued fraction expansion and regular continued fraction expansion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14534",
  "author_total": 2,
  "authors": [
    {
      "name": "Cheng Liu"
    },
    {
      "name": "Lingmin Liao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6ghpjkqa7xa80.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14534",
  "fulltext_url": null,
  "id": "6ghpjkqa7xa80",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14534",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7221759248",
  "pdf_url": "https://arxiv.org/pdf/2610.14534",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6ghpjkqa7xa80.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14534",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14534"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Survivor Set of Gauss Map with a Hole at $1$",
  "title_text": "Survivor Set of Gauss Map with a Hole at $1$",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/6ghpjkqa7xa80",
  "venue": "arXiv",
  "versions": []
}