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

TL;DR

This paper proves that periodic points of holomorphic endomorphisms of projective space distribute exponentially fast towards the equilibrium measure as their period increases, extending previous results to higher dimensions.

Contribution

It establishes exponential equidistribution of periodic points for endomorphisms of projective space, generalizing known results from dimension one to higher dimensions.

Findings

Periodic points equidistribute exponentially fast towards the equilibrium measure.

Existence of many periodic cycles with large multipliers in the small Julia set.

Quantification of Lyubich's theorem for higher dimensions.

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.