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Cyclotomic integral points for affine dynamics

Zhuchao Ji, Junyi Xie, Geng-Rui Zhang

TL;DR

This work develops a higher-dimensional rigidity theory for algebraic dynamical systems by linking the distribution of cyclotomic preperiodic points to monomial-type dynamics on tori. The authors introduce three conditions—dense cyclotomic integral points (DCI), bounded height (BH), and almost invariant (AI)—and prove that, under these hypotheses, a dominant endomorphism on an affine variety must, after an iterate, factor through a surjective endomorphism of a linear torus, i.e., it is of monomial type (and strongly monomial type under cohomological hyperbolicity). The paper then extends these results to backward orbits and to periodic points of Hénon-type automorphisms, yielding rigidity and non-density statements for cyclotomic points in broader dynamical settings. By combining Loxton-type decompositions, torsion-point theory on tori, and dynamical-degree analysis, the results generalize the one-variable DZ07 theorem to higher dimensions and illuminate the role of torus dynamics in unlikely intersection phenomena.

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 $f$-preperiodic cyclotomic 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 is a higher-dimensional generalization of a theorem of Dvornicich and Zannier on cyclotomic preperiodic points of one-variable polynomials. In fact, we prove a much more general rigidity result for all dominant endomorphisms $f$ on an affine variety $X$ defined over a number field, regarding "almost $f$-invariant" Zariski-dense subsets of cyclotomic integral points. As applications, we also 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 Hénon type on $\mathbb{A}^N$.

Cyclotomic integral points for affine dynamics

TL;DR

This work develops a higher-dimensional rigidity theory for algebraic dynamical systems by linking the distribution of cyclotomic preperiodic points to monomial-type dynamics on tori. The authors introduce three conditions—dense cyclotomic integral points (DCI), bounded height (BH), and almost invariant (AI)—and prove that, under these hypotheses, a dominant endomorphism on an affine variety must, after an iterate, factor through a surjective endomorphism of a linear torus, i.e., it is of monomial type (and strongly monomial type under cohomological hyperbolicity). The paper then extends these results to backward orbits and to periodic points of Hénon-type automorphisms, yielding rigidity and non-density statements for cyclotomic points in broader dynamical settings. By combining Loxton-type decompositions, torsion-point theory on tori, and dynamical-degree analysis, the results generalize the one-variable DZ07 theorem to higher dimensions and illuminate the role of torus dynamics in unlikely intersection phenomena.

Abstract

Let be a regular endomorphism of algebraic degree (i.e., extends to an endomorphism on of algebraic degree ) defined over a number field. We prove that if the set of -preperiodic cyclotomic points is Zariski-dense in , then some iterate () is a quotient of a surjective algebraic group endomorphism , over . This is a higher-dimensional generalization of a theorem of Dvornicich and Zannier on cyclotomic preperiodic points of one-variable polynomials. In fact, we prove a much more general rigidity result for all dominant endomorphisms on an affine variety defined over a number field, regarding "almost -invariant" Zariski-dense subsets of cyclotomic integral points. As applications, we also apply our results to backward orbits of regular endomorphisms on of algebraic degree , and to periodic points of automorphisms of Hénon type on .

Paper Structure

This paper contains 11 sections, 14 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Applications
  4. Motivation and previous results
  5. Sketch of the proof
  6. Preliminaries
  7. Cyclotomic extension and Loxton theorem
  8. Linear tori and torsion points theorem
  9. Dynamical degrees
  10. Proof of the main results
  11. Proofs of the applications

Key Result

Theorem 1.2

Let $X$ be a geometrically irreducible affine variety of dimension $d\geq1$ and $f:X\to X$ a dominant endomorphism, both defined over a number field $K$. Fix an embedding $X\subseteq \mathbb A^N_K$ with coordinates $(x_1,\dots,x_N)$ of $\mathbb A^N$. Assume that $P$ is a subset of $X(\overline{K})$ Then $(X,f)$ is of monomial type.

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9: Dvornicich-Zannier
  • Remark 1.10
  • ...and 15 more