Cyclotomic integral points for affine dynamics

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 $\mathbb{A}^N$.