A general endomorphism of ℙk has trivial iterated centralizer
Abstract
Fix integers k,d≥2. Let Enddk denote the parameter space of endomorphisms of ℙk of algebraic degree d. We prove that there exists a dense Zariski open subset Ud,k⊂Enddk, defined over ℚ, such that, for every f∈ Ud,k(ℂ), every dominant rational self-map of ℙk commuting with an iterate of f is itself an iterate of f. The key intermediate result classifies dominant rational semiconjugacies to iterates of f from normal projective varieties of dimension k. The proof uses finite-level monodromy and its action on the rooted preimage tree of a point outside the branch locus of an iterate of f. An analogous centralizer theorem holds generically for regular polynomial endomorphisms of ℂk. These results extend several theorems of Pakovich to higher dimensions. We give two applications. We first classify periodic dominant correspondences between a general endomorphism and an arbitrary endomorphism of degree at least two. We then show that, for a general f∈Enddk, an irreducible hypersurface of ℙk is f-special in the sense of Ghioca--Tucker and DeMarco--Mavraki if and only if it is f-preperiodic.
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BibTeX
@misc{zhang2026general,
title = {{A general endomorphism of $\mathbbP^k$ has trivial iterated centralizer}},
author = {Yugang Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09858},
eprint = {2609.09858},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09858},
}
RIS
TY - UNPB
TI - A general endomorphism of ℙ^k has trivial iterated centralizer
AU - Yugang Zhang
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.09858
UR - https://arxiv.org/abs/2609.09858
AB - Fix integers k,d≥2. Let End_d^k denote the parameter space of endomorphisms of ℙ^k of algebraic degree d. We prove that there exists a dense Zariski open subset U_d,k⊂End_d^k, defined over ℚ, such that, for every f∈ U_d,k(ℂ), every dominant rational self-map of ℙ^k commuting with an iterate of f is itself an iterate of f. The key intermediate result classifies dominant rational semiconjugacies to iterates of f from normal projective varieties of dimension k. The proof uses finite-level monodromy and its action on the rooted preimage tree of a point outside the branch locus of an iterate of f. An analogous centralizer theorem holds generically for regular polynomial endomorphisms of ℂ^k. These results extend several theorems of Pakovich to higher dimensions. We give two applications. We first classify periodic dominant correspondences between a general endomorphism and an arbitrary endomorphism of degree at least two. We then show that, for a general f∈End_d^k, an irreducible hypersurface of ℙ^k is f-special in the sense of Ghioca--Tucker and DeMarco--Mavraki if and only if it is f-preperiodic.
ER -