Lower Bounds for Galois Orbits of periodic points for polarized endomorphisms

TL;DR

This paper establishes exponential lower bounds on the Galois orbits of periodic points for polarized endomorphisms on higher-dimensional varieties and provides a quantitative rate of equidistribution towards the equilibrium measure.

Contribution

It extends previous results to higher dimensions and introduces quantitative bounds and equidistribution rates for periodic points under polarized endomorphisms.

Findings

Exponential lower bounds on Galois degrees of periodic points.

Quantitative rate of equidistribution to the equilibrium measure.

Extension of prior one-dimensional results to higher dimensions.

Abstract

Let $K$ be a number field, $X$ a smooth projective variety over $K$ and $f: X \to X$ a polarized endomorphism of degree $d \geq 2$. We prove an exponential lower bound on $[K(\Per_n):K]$, where $\Per_n$ is the set of $n$-periodic points, extending results of [Yap24] to higher dimensions. We also prove a quantitative rate of equidistribution for $\Per_n$ to the equilibrium measure.