{
  "abstract": "Let $Γ<\\operatorname{PSL}_2(\\mathbb R)$ be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let $ρ:Γ\\to G$ be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing for Bowen--Margulis--Sullivan measures on $ρ(Γ)\\backslash G/M$, where $M$ is the centralizer of a maximal split torus in a compatible maximal compact subgroup. For every $u$ in the interior of the limit cone of $ρ(Γ)$, correlations of smooth compactly supported functions along $\\exp(tu)$, normalized by $t^{(\\operatorname{rank}_{\\mathbb R}G-1)/2}$, admit a complete asymptotic expansion in integer powers of $t^{-1}$. We also obtain a uniform Gaussian asymptotic for transverse displacements, with an integrable remainder. We deduce asymptotics for matrix coefficients of the quasi-regular representation on $L^2(ρ(Γ)\\backslash G)$ and for orbital counting in Riemannian balls. More generally, these conclusions hold for Zariski dense, relatively Borel Anosov representations of $Γ$ satisfying a peripheral root-balance condition. This includes relatively Borel Anosov representations into split simple groups of rank at most three and self-joinings of type-preserving geometrically finite real hyperbolic representations. To prove these results, we develop an intrinsic cusp geometry on the limit set of $ρ(Γ)$ in the Furstenberg boundary and use it to construct a countable Markov coding. We establish exponential moment bounds and uniform distortion estimates for the unbounded vector-valued return cocycle. Using peripheral Cartan growth and Zariski density, we establish the quantitative non-integrability needed for the spectral analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12254",
  "author_total": 3,
  "authors": [
    {
      "name": "Dongryul M. Kim"
    },
    {
      "name": "Hee Oh"
    },
    {
      "name": "Wenyu Pan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5bhfzft367f26.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12254",
  "fulltext_url": null,
  "id": "5bhfzft367f26",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12254",
  "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"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7221081158",
  "pdf_url": "https://arxiv.org/pdf/2610.12254",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5bhfzft367f26.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12254",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12254"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Higher-rank local mixing for cusped positive representations",
  "title_text": "Higher-rank local mixing for cusped positive representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/5bhfzft367f26",
  "venue": "arXiv",
  "versions": []
}