{
  "abstract": "Let $H\\in C^\\infty(\\mathbb R^n\\times\\mathbb T^n)$ be a periodic Tonelli Hamiltonian with critical value $c$. For each $k\\in\\mathbb N$, let $u_k$ be the normalized minimizer of the variational functional introduced by Evans[7], \\[ I_k[w]=\\int_{\\mathbb T^n} e^{kH(Dw,x)}\\,dx, \\qquad \\int_{\\mathbb T^n}w\\,dx=0. \\] If $u_\\infty$ is a uniform limit of a subsequence of $\\{u_k\\}$ and the Mather quotient $({A}_M,δ_M)$ satisfies $H^1( A_M,δ_M)=0$, then $u_\\infty$ is a critical subsolution that is strict outside ${A}$ and \\[ {A} = \\{x\\in\\mathbb T^n\\,:\\,Du_\\infty(x)\\ \\text{exists and }H(Du_\\infty(x),x)=c\\}=\\{x\\in\\mathbb T^n\\,:\\,u_\\infty(x)=u_{-}(x)\\}, \\] where ${A}$ is the projected Aubry set and $u_{-}$ is the backward weak KAM solution associated with $u_\\infty$. In particular, by the theorem of Fathi--Figalli--Rifford[10], this conclusion holds for all smooth Tonelli Hamiltonians on $\\mathbb T^n$ when $n\\leq3$. This characterization also suggests a natural numerical localization principle for approximating the entire Aubry set through near-contact sets between $u_k$ and its large-time backward Lax--Oleinik evolution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01557",
  "author_total": 2,
  "authors": [
    {
      "name": "Hung V. Tran"
    },
    {
      "name": "Yifeng Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fcy7ncdt5g32m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01557",
  "fulltext_url": null,
  "id": "fcy7ncdt5g32m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01557",
  "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.AP"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7206160120",
  "pdf_url": "https://arxiv.org/pdf/2609.01557",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fcy7ncdt5g32m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01557",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01557"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "$L^\\infty$ Variational Approximation of the Aubry Set",
  "title_text": "L^∞ Variational Approximation of the Aubry Set",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/fcy7ncdt5g32m",
  "venue": "arXiv",
  "versions": []
}