{
  "abstract": "We show that finite-order exponential mixing does not imply the central limit theorem for smooth diffeomorphisms. For every integer $R\\geq2$, we construct a $\\mathcal{C}^{\\infty}$ diffeomorphism $T:M\\to M$ of a compact smooth manifold $M$ and a $T$-invariant probability measure $μ$ such that $(T,μ)$ is exponentially mixing of order $R$ for $\\mathcal{C}^1(M)$ observables, but not of order $2R+2$. For some centred observable $f_0\\in\\mathcal{C}^{\\infty}(M)$, no subsequence of the normalised sums $\\frac{1}{\\sqrt n}\\sum_{i=0}^{n-1}f_0\\circ T^i$ converges in distribution to a Gaussian law. In particular, exponential decay of two-point correlations implies neither exponential mixing of all orders nor the central limit theorem. We realise $(T,μ)$ in two ways. In the first, $T$ is a fixed linear Anosov diffeomorphism of $\\mathbb T^2$ and $μ$ is singular with respect to Lebesgue measure. In the second, $M$ can be any compact smooth surface without boundary and $μ$ is a physical measure whose statistical basin contains a nonempty open set. The main theorems have been formalised in Lean 4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17162",
  "author_total": 2,
  "authors": [
    {
      "name": "Bernat Bassols Cornudella"
    },
    {
      "name": "Matheus M Castro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ax0mcat8g8z9c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17162",
  "fulltext_url": null,
  "id": "ax0mcat8g8z9c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17162",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7221792440",
  "pdf_url": "https://arxiv.org/pdf/2610.17162",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ax0mcat8g8z9c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17162",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17162"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Finite-order exponential mixing does not imply the central limit theorem",
  "title_text": "Finite-order exponential mixing does not imply the central limit theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/ax0mcat8g8z9c",
  "venue": "arXiv",
  "versions": []
}