Degree growth, orbit graphs, and functoriality for birational dynamical systems
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.
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BibTeX
@misc{takenawa2026degree,
title = {{Degree growth, orbit graphs, and functoriality for birational dynamical systems}},
author = {Tomoyuki Takenawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.29274},
eprint = {2606.29274},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.29274},
}
RIS
TY - UNPB
TI - Degree growth, orbit graphs, and functoriality for birational dynamical systems
AU - Tomoyuki Takenawa
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2606.29274
UR - https://arxiv.org/abs/2606.29274
AB - 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.
ER -