{
  "abstract": "Let $H\\in C^2(T^*M)$ be a Tonelli Hamiltonian on a closed connected manifold and let $u_λ$ solve \\[ λu_λ+H(x,Du_λ)=c(H)\\qquad\\text{in }M. \\] We study the convergence rate of $u_λ$ to the selected critical solution $u_0$. Assume that the lifted Aubry set is a finite union $\\widetilde{A}=Γ_1\\sqcup\\cdots\\sqcupΓ_N$, where each $Γ_i$ is either a hyperbolic equilibrium or a periodic orbit hyperbolic in the critical energy level. We prove \\[ -Cλ\\le u_λ-u_0\\le Cλ|\\logλ|. \\] Let $μ_i$ be the projected Mather measure associated with $Γ_i$. If \\[ \\int_M u_0\\,\\mathrm dμ_i=0\\qquad \\text{for every }i, \\] then \\[ \\|u_λ-u_0\\|_\\infty\\le Cλ. \\] In particular, if the lifted Aubry set consists of a single hyperbolic equilibrium or a single hyperbolic periodic orbit, the convergence rate is $O(λ)$. We give examples showing that both convergence rates $O(λ)$ and $O(λ|\\logλ|)$ are optimal. Without hyperbolicity, finite-order degenerate examples give lower bounds of order $λ^{1/(2r-1)}$ with $r\\ge2$. We also construct infinite-order degenerate examples with arbitrarily slow convergence. Taken together, these results provide, to our knowledge, the first systematic quantitative theory for the vanishing discount problem in the Tonelli setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02779",
  "author_total": 1,
  "authors": [
    {
      "name": "Panrui Ni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aq4jzsty41w28.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02779",
  "fulltext_url": null,
  "id": "aq4jzsty41w28",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02779",
  "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",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7207749366",
  "pdf_url": "https://arxiv.org/pdf/2609.02779",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aq4jzsty41w28.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02779",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02779"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Sharp convergence rates for the vanishing discount problem with hyperbolic Aubry sets",
  "title_text": "Sharp convergence rates for the vanishing discount problem with hyperbolic Aubry sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/aq4jzsty41w28",
  "venue": "arXiv",
  "versions": []
}