{
  "abstract": "We study the topological entropy of saturated sets for continuous flows, with emphasis on singular hyperbolic dynamics. For a $C^r$-generic ($r \\geq 2$) vector field $X$ carrying a geometric Lorenz attractor $Λ$, we prove that for every nonempty compact connected set $\\mathcal{K} \\subseteq \\mathcal{M}(Φ,Λ)$ and every open set $U \\subseteq M$ with $U \\cap Λ\\neq \\emptyset$, $h_{\\mathrm{top}}(Φ,G_{\\mathcal{K}}(Φ) \\cap \\operatorname{Trans}(Φ,Λ) \\cap U)=\\inf_{μ\\in \\mathcal{K}} h_μ(ϕ_1)$, where $Φ=(ϕ_t)_{t \\in \\mathbb{R}}$ is the flow induced by $X$ and $\\operatorname{Trans}(Φ,Λ)$ denotes the set of points whose positive orbits are dense in $Λ$. The same formula holds $C^1$-generically for singular hyperbolic attractors. In contrast with the uniformly hyperbolic case, we also prove that, for vector fields in a dense subset of $\\mathcal{X}^r(M^3)$, geometric Lorenz attractors admit a positive-entropy invariant measure with no generic points. An analogous conclusion holds for vector fields in a dense subset of $\\mathcal{X}^1(M)$ whose non-trivial singular hyperbolic attractors contain singularities and satisfy $\\dim E^{cu}=2$. These examples show that the discrete-time saturated entropy theory does not extend directly to singular flows.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18148",
  "author_total": 4,
  "authors": [
    {
      "name": "Zheng Huang"
    },
    {
      "name": "Xueting Tian"
    },
    {
      "name": "Xiaodong Wang"
    },
    {
      "name": "Xutong Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cb0n43y8w59a1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18148",
  "fulltext_url": null,
  "id": "cb0n43y8w59a1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18148",
  "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": "W7221779682",
  "pdf_url": "https://arxiv.org/pdf/2610.18148",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cb0n43y8w59a1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18148",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18148"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Topological entropy of saturated sets for Lorenz attractors",
  "title_text": "Topological entropy of saturated sets for Lorenz attractors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/cb0n43y8w59a1",
  "venue": "arXiv",
  "versions": []
}