{
  "abstract": "The degrees of the iterates of a projective endomorphism grow in two layers: an exponential rate measured by the dynamical degrees, and a polynomial correction carried by the peripheral Jordan blocks. Log-concavity constrains the first layer; we show that the Hodge index theorem already constrains the second, in every codimension and in arbitrary characteristic. Let $f$ be a surjective endomorphism of a normal projective $d$-fold over an algebraically closed field, with dynamical degrees $λ_i$, so that $\\textrm{deg}_i(f^n)\\asympλ_i^n n^{μ_i}$ for a unique integer $μ_i\\ge0$. We prove that these exponents are coupled across three adjacent codimensions: strict log-concavity of $(λ_i)_i$ at $i$ forces $μ_i=0$, while equality gives \\[ 0\\ \\le\\ 2μ_i-μ_{i-1}-μ_{i+1}\\ \\le\\ 4 . \\] Consequently, $μ_i\\le2i(d-i)$, and this is sharp for every $i$. For a zero-entropy automorphism, we prove that every mixed orbit function indexed by any weak composition $γ$ of $d$ is a multivariate quasipolynomial of even total degree at most $d^2-\\sum_jγ_j^2$. For a zero-entropy holomorphic automorphism $g$ of a compact Kähler $d$-fold $Y$, the same method gives \\[\\left\\|(g^n)^*|_{H^{p,q}(Y,\\mathbb{C})}\\right\\| =O\\bigl(n^{p(d-p)+q(d-q)}\\bigr).\\] Powers of elliptic curves attain all exponent bounds as well as the constant $4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29877",
  "author_total": 2,
  "authors": [
    {
      "name": "Fei Hu"
    },
    {
      "name": "Chen Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cr3qzh7tvxnn6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29877",
  "fulltext_url": null,
  "id": "cr3qzh7tvxnn6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29877",
  "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.AG"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214315710",
  "pdf_url": "https://arxiv.org/pdf/2609.29877",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cr3qzh7tvxnn6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29877",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29877"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Uniform Turán estimates and sharp bounds for degree and mixed orbit growth",
  "title_text": "Uniform Turán estimates and sharp bounds for degree and mixed orbit growth",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/cr3qzh7tvxnn6",
  "venue": "arXiv",
  "versions": []
}