Cyclotomic integral points for affine dynamics

TL;DR

This paper proves that if cyclotomic preperiodic points are Zariski-dense for a higher-dimensional affine endomorphism, then some iterate is a quotient of a group endomorphism, extending known results from one-variable polynomials.

Contribution

It generalizes a theorem on cyclotomic preperiodic points from one-variable polynomials to higher dimensions and establishes a rigidity result for dominant endomorphisms on affine varieties.

Findings

Zariski-density implies some iterate is a quotient of a group endomorphism

Extends Dvornicich and Zannier's theorem to higher dimensions

Applies to backward orbits and automorphisms of Hénon type

Abstract

Let $f:\mathbb{A}^N\to\mathbb{A}^N$ be a regular endomorphism of algebraic degree $d\geq2$ (i.e., $f$ extends to an endomorphism on $\mathbb{P}^N$ of algebraic degree $d$) defined over a number field. We prove that if the set of cyclotomic $f$-preperiodic points is Zariski-dense in $\mathbb{A}^N$, then some iterate $f^{\circ l}$ ($l\geq1$) is a quotient of a surjective algebraic group endomorphism $g:\mathbb{G}_m^N\to\mathbb{G}_m^N$, over $\overline{\mathbb{Q}}$. This result generalizes a theorem of Dvornicich and Zannier on cyclotomic preperiodic points of one-variable polynomials to higher dimensions. In fact, we prove a much more general rigidity result for dominant endomorphisms $f$ on an affine variety $X$ defined over a number field, concerning "almost $f$-invariant" Zariski-dense subsets of cyclotomic integral points. We apply our results to backward orbits of regular endomorphisms on $\mathbb{A}^N$ of algebraic degree $d\geq2$, and to periodic points of automorphisms of H\'enon type on $\mathbb{A}^N$.