{
  "abstract": "In this paper, we show that for any $S$-unimodal map $T$ on $[0,1]$ with a non-flat critical point the Lyapunov exponent exists for Lebesgue almost every point and is equal to a constant $λ_T$. Moreover, $λ_T=0$ if and only if $T$ admits neither an absolutely continuous $T$-invariant probability measure with positive entropy nor a strictly stable periodic orbit. Consequently, if an $S$-unimodal map with a non-flat critical point is infinitely renormalizable or non-statistical then for Lebesgue almost every $x\\in [0,1]$ the Lyapunov exponent along the orbit of $x$ exists and is equal to $0$. A key ingredient is the following result of independent interest. If an $S$-unimodal map with a non-flat critical point has no periodic attractor then for Lebesgue almost every $x\\in [0,1]$ the lower Lyapunov exponent along the orbit of $x$ is non-negative. This shows that, in the absence of periodic attractors, exponential contraction cannot occur along the orbit of Lebesgue almost every point.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04187",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuya Arima"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3ph1vn96jsh1c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.04187",
  "fulltext_url": null,
  "id": "3ph1vn96jsh1c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04187",
  "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": "W7167637127",
  "pdf_url": "https://arxiv.org/pdf/2607.04187",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3ph1vn96jsh1c.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2607.04187",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04187"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Existence of the Lyapunov exponent for $S$-unimodal maps",
  "title_text": "Existence of the Lyapunov exponent for S-unimodal maps",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:25Z",
  "url": "https://research.draconitic.com/paper/3ph1vn96jsh1c",
  "venue": "arXiv",
  "versions": []
}