{
  "abstract": "We study the question of existence and multiplicity of homogeneous Markov Gibbs measures on Cayley trees with spin space $\\mathbb{Z}$. We prove that, under a natural summability assumption on the interaction function, there are always at least two homogeneous Markov Gibbs measures satisfying two natural symmetry conditions. Moreover, any two homogeneous Markov Gibbs measures satisfying a suitable non-degeneracy assumption can be superimposed and produce infinitely many other homogeneous Markov Gibbs measures. When the interaction function $Q$ has bounded support in $\\mathbb{Z}$, there is a one-to-one correspondence between homogeneous Markov Gibbs measures and suitable homoclinic orbits of a symplectomorphism of $\\mathbb{R}^{2n}$, where $n$ is the smallest natural number such that $Q(i)=0$ for every $i>n$. For $n=1$, results from symbolic dynamics and horseshoes imply the existence of homogeneous Markov Gibbs measures whose single-site marginal is concentrated near an arbitrary finite subset of $N\\mathbb{Z}$, where $N$ is a sufficiently large integer. In the case of interaction functions with infinite support, there is a one-to-one correspondence between homogeneous Markov Gibbs measures and suitable homoclinic orbits of a Lagrangian correspondence on the cotangent bundle of a Banach space.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33287",
  "author_total": 2,
  "authors": [
    {
      "name": "Alberto Abbondandolo"
    },
    {
      "name": "Pietro Majer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4qm866t7q3rr9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33287",
  "fulltext_url": null,
  "id": "4qm866t7q3rr9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33287",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7224020388",
  "pdf_url": "https://arxiv.org/pdf/2610.33287",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4qm866t7q3rr9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33287",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33287"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Gibbs measures on Cayley trees and homoclinic orbits",
  "title_text": "Gibbs measures on Cayley trees and homoclinic orbits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/4qm866t7q3rr9",
  "venue": "arXiv",
  "versions": []
}