{
  "abstract": "Let $X^t$ be a volume-preserving Anosov flow of class $C^7$ on a closed $3$-manifold. We consider a sequence of periodic orbits which shadow a homoclinic loop of a fixed periodic orbit $γ$, and derive higher-order asymptotic expansions for their periods $T_n$ and for the logarithms of their unstable multipliers $\\logλ_n$. We give geometric interpretations of the coefficients in these expansions, in particular in terms of the Anosov cocycle (à la Hurder-Katok) and the longitudinal Anosov cocycle (à la Foulon-Hasselblatt), evaluated on $γ$. Following the idea of De Simoi-Leguil-Vinhage-Yang, we compare these asymptotics when volume is the measure of maximal entropy. In this case, the equilibrium-state characterizations of volume and the measure of maximal entropy synchronize periods and exponents: $\\logλ_n=hT_n$. Matching the coefficients of the two asymptotic expansions forces the cohomology classes of both cocycles to vanish. Then the work of Foulon-Hasselblatt on the longitudinal Anosov cocycle, together with Foulon's work on the contact case, yields the Foulon Conjecture for flows of class $C^7$: if volume is the measure of maximal entropy, then, up to a finite cover, $X^t$ is smoothly conjugate either to a constant-roof suspension of a hyperbolic automorphism of the $2$-torus or to a finite cover of the constant-speed geodesic flow on a closed hyperbolic surface.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31823",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrey Gogolev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7rm2wj22wwjyb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31823",
  "fulltext_url": null,
  "id": "7rm2wj22wwjyb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31823",
  "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"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7223702575",
  "pdf_url": "https://arxiv.org/pdf/2610.31823",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7rm2wj22wwjyb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.31823",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31823"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Higher-order asymptotic formulae for periodic orbits of 3-dimensional volume-preserving Anosov flows and the Foulon conjecture",
  "title_text": "Higher-order asymptotic formulae for periodic orbits of 3-dimensional volume-preserving Anosov flows and the Foulon conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/7rm2wj22wwjyb",
  "venue": "arXiv",
  "versions": []
}