{
  "abstract": "In this paper we study one-parameter families of exact standard maps of the form $$\\widehat{f}_{k,g}(\\widehat{x},\\widehat{y}) =\\bigl(\\widehat{x}+\\widehat{y}+kg(\\widehat{x}) \\bmod 1,\\widehat{y}+kg(\\widehat{x})\\bigr),$$ where $k$ is a real parameter and $g:\\mathbb{T} \\rightarrow \\mathbb{R}$ is real-analytic, 1-periodic and has zero mean. The most famous example is when $g(\\widehat{x})=\\frac{\\sin(2π\\widehat{x})}{2π}$. In this case, the closed subset $A_{\\sin(2π\\cdot)/(2π)}\\subset \\mathbb{R}$ of values of $k$ for which $\\widehat{f}_{k,\\sin(2π\\cdot)/(2π)}$ has rotational invariant curves is conjectured to be a closed interval. Much has been done towards this conjecture, both numerically and analytically. But a complete proof has not yet been achieved. Our approach here goes in the opposite direction: we construct a function $θ:\\mathbb{T} \\rightarrow \\mathbb{R}$ satisfying the above conditions, such that $A_θ$ is not an interval: It contains at least 2 different connected components, which are non-degenerate closed intervals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28449",
  "author_total": 1,
  "authors": [
    {
      "name": "Salvador Addas-Zanata"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d94xbxgf61s4z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28449",
  "fulltext_url": null,
  "id": "d94xbxgf61s4z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28449",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7223126523",
  "pdf_url": "https://arxiv.org/pdf/2610.28449",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d94xbxgf61s4z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28449",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28449"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Real-analytic standard families and rotational invariant curves",
  "title_text": "Real-analytic standard families and rotational invariant curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/d94xbxgf61s4z",
  "venue": "arXiv",
  "versions": []
}