Exponential equidistribution of periodic points for endomorphisms of $\mathbb P^k$

Henry de Thélin; Tien-Cuong Dinh; Lucas Kaufmann

Exponential equidistribution of periodic points for endomorphisms of $\mathbb P^k$

Dynamical Systems 2025-05-02 v2 Complex Variables

Authors: Henry de Thélin , Tien-Cuong Dinh , Lucas Kaufmann

Abstract

Let $f$ be a holomorphic endomorphism of $\mathbb P^k$ of algebraic degree $d\geq 2$ . We show that the periodic points of $f$ of period $n$ equidistribute towards the equilibrium measure of $f$ exponentially fast as $n$ tends to infinity. This quantifies a theorem of Lyubich for $k=1$ and of Briend-Duval for $k\geq 2$ . A byproduct of our proof is the existence of a large number of periodic cycles in the small Julia set with large multipliers.

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Cite

@article{arxiv.2409.19787,
  title  = {Exponential equidistribution of periodic points for endomorphisms of $\mathbb P^k$},
  author = {Henry de Thélin and Tien-Cuong Dinh and Lucas Kaufmann},
  journal= {arXiv preprint arXiv:2409.19787},
  year   = {2025}
}

Comments

In this version, with an additional coauthor, we prove that for any holomorphic endomorphism of P^k, the repelling periodic points in the small Julia set converge exponentially fast to the equilibrium measure. The proof is different and independent of the previous one when k=1

Exponential equidistribution of periodic points for endomorphisms of $\mathbb P^k$

Abstract

Keywords

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