{
  "abstract": "Mather's $β$-function associates with each rotation number the least average action carried by an orbit of a twist map. For analytic standard maps in the KAM regime, we prove the following. For any family of $d \\ge 1$ directions of perturbation satisfying a natural symmetry condition, the jet of the $β$-function at a single algebraic Diophantine rotation number locally determines the potential for an open and prevalent, hence dense, set of base potentials in the KAM domain. In particular, we obtain that for a residual and prevalent set of even potentials in the KAM domain, every real analytic deformation with finite Fourier support that preserves the full jet of the $β$-function at a fixed algebraic Diophantine rotation number is constant. These results are non perturbative within the KAM domain. As an intermediate result of independent interest, we establish a KAM theorem giving joint $C^\\infty$--holomorphic dependence of the invariant curves, and hence of the $β$-function, on complex holomorphic potentials and on a domain of complexified rotation numbers whose boundary contains real Diophantine numbers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15617",
  "author_total": 1,
  "authors": [
    {
      "name": "Mathieu Helfter"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d0h4pgpv74b24.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15617",
  "fulltext_url": null,
  "id": "d0h4pgpv74b24",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15617",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.15617",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d0h4pgpv74b24.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15617",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15617"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Jet rigidity of Mather's $β$-function and complexified KAM curves holomorphic in potential for analytic standard maps",
  "title_text": "Jet rigidity of Mather's β-function and complexified KAM curves holomorphic in potential for analytic standard maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/d0h4pgpv74b24",
  "venue": "arXiv",
  "versions": []
}