Uniform Turán estimates and sharp bounds for degree and mixed orbit growth
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 λi, so that degi(fn)asympλin nμi for a unique integer μi≥0. We prove that these exponents are coupled across three adjacent codimensions: strict log-concavity of (λi)i at i forces μi=0, while equality gives 0 ≤ 2μi−μi−1−μi+1 ≤ 4 . Consequently, μi≤2i(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 d2−∑jγj2. For a zero-entropy holomorphic automorphism g of a compact Kähler d-fold Y, the same method gives |(gn)*|Hp,q(Y,ℂ)| =O(np(d−p)+q(d−q)). Powers of elliptic curves attain all exponent bounds as well as the constant $4$.
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BibTeX
@misc{hu2026uniform,
title = {{Uniform Turán estimates and sharp bounds for degree and mixed orbit growth}},
author = {Fei Hu and Chen Jiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29877},
eprint = {2609.29877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29877},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniform Turán estimates and sharp bounds for degree and mixed orbit growth
AU - Fei Hu
AU - Chen Jiang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29877
UR - https://arxiv.org/abs/2609.29877
AB - 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 deg_i(f^n)asympλ_i^n n^μ_i for a unique integer μ_i≥0. 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 ≤ 2μ_i−μ_i−1−μ_i+1 ≤ 4 . Consequently, μ_i≤2i(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−∑_jγ_j^2. For a zero-entropy holomorphic automorphism g of a compact Kähler d-fold Y, the same method gives |(g^n)^*|_H^p,q(Y,ℂ)| =O(n^p(d−p)+q(d−q)). Powers of elliptic curves attain all exponent bounds as well as the constant $4$.
ER -