{
  "abstract": "The purpose of this paper is to give a natural divisor-theoretic formulation of the counting method introduced by Halburd for computing degree growth, in a form applicable to birational dynamical systems on varieties of arbitrary dimension. Instead of counting only preimages of special values, we follow time-indexed divisorial conditions through singularity patterns. These conditions are recorded on normalized finite-window orbit graphs, where the relevant multiplicities are realized as divisorial valuations of pullbacks of time-indexed divisors. This construction explains how the elementary computations appearing in singularity patterns can be interpreted as degree relations on a single normal variety. We then show that further relations arise from the failure of functoriality of pullbacks: when the center of a divisor enters the relevant indeterminacy locus, a degree-drop divisor appears. Under suitable finite-type assumptions, the two kinds of relations lead to closed linear difference systems governing degree sequences. Several examples, including higher-dimensional ones, demonstrate that the two mechanisms are complementary and that their combination determines the degree growth in cases where either mechanism alone is insufficient.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.29274",
  "author_total": 1,
  "authors": [
    {
      "name": "Tomoyuki Takenawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8etb3mc9xk6e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.29274",
  "fulltext_url": null,
  "id": "e8etb3mc9xk6e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.29274",
  "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.AG"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7166702876",
  "pdf_url": "https://arxiv.org/pdf/2606.29274",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8etb3mc9xk6e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2606.29274",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.29274"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Degree growth, orbit graphs, and functoriality for birational dynamical systems",
  "title_text": "Degree growth, orbit graphs, and functoriality for birational dynamical systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/e8etb3mc9xk6e",
  "venue": "arXiv",
  "versions": []
}