Lower Bounds for Galois Orbits of periodic points for polarized endomorphisms

TL;DR

The paper extends Yap24’s results on Galois lower bounds and equidistribution of periodic points from surfaces to higher-dimensional polarized endomorphisms. It develops a robust framework based on density of positive closed currents to bound intersections and to derive exponential growth in Galois orbits of periodic points, while establishing a quantitative rate of equidistribution toward the equilibrium measure at archimedean places. A parallel analysis for negative orbits uses multiplicities and the maximal totally invariant set $\mathcal{E}$ to obtain analogous growth results. The work combines intersection theory, height machinery, and complex dynamics to yield dimension-agnostic bounds and rates, broadening the Diophantine-dynamics toolkit for polarized endomorphisms. This has potential implications for understanding arithmetic properties of higher-dimensional dynamical systems and their equidistribution behavior across places.

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.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: see DS18
• Remark 2.2
• Example 2.3: good local model
• Corollary 2.4
• Proposition 3.1
• Remark 3.2
• Theorem 4.1