PreprintCC BY 4.0arXiv
Height growth of rational self-maps
Abstract
Let K be a number field or a function field of a curve. Let f be a dominant rational self-map of a normal projective variety X of dimension d over K, and let hH≥1 be an ample Weil height. Let λ1(f) denote the first dynamical degree of f. We show that there is a constant Cf,hH>0 such that, for any n≥1 and any point P∈ X(K̅) with well-defined forward orbit, hH(fn(P))≤ Cf,hHhH(P)n2(d−1)λ1(f)n if λ1(f)>1, and hH(fn(P))≤ Cf,hHhH(P)n2d if λ1(f)=1. Over number fields, the proof combines a new uniform upper bound for arithmetic degrees with a uniform comparison of point heights. Over function fields, it relies on the corresponding uniform upper bound for geometric degrees on a projective model.
Related
- Degree growth of rational and meromorphic self-maps
- Higher arithmetic degrees of dominant rational self-maps
- Degree Growth of Rational Maps with Laurent Property
- Height functions on singular surfaces parameterized by smooth maps 𝒜-equivalent to Hk
- Smooth maps onto regions in Euclidean spaces respecting height functions
BibTeX
@misc{hu2026height,
title = {{Height growth of rational self-maps}},
author = {Fei Hu and Jiarui Song},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33570},
eprint = {2610.33570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33570},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Height growth of rational self-maps
AU - Fei Hu
AU - Jiarui Song
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33570
UR - https://arxiv.org/abs/2610.33570
AB - Let K be a number field or a function field of a curve. Let f be a dominant rational self-map of a normal projective variety X of dimension d over K, and let h_H≥1 be an ample Weil height. Let λ_1(f) denote the first dynamical degree of f. We show that there is a constant C_f,h_H>0 such that, for any n≥1 and any point P∈ X(K̅) with well-defined forward orbit, h_H(f^n(P))≤ C_f,h_Hh_H(P)n^2(d−1)λ_1(f)^n if λ_1(f)>1, and h_H(f^n(P))≤ C_f,h_Hh_H(P)n^2d if λ_1(f)=1. Over number fields, the proof combines a new uniform upper bound for arithmetic degrees with a uniform comparison of point heights. Over function fields, it relies on the corresponding uniform upper bound for geometric degrees on a projective model.
ER -