Algebraicity criteria, invariant subvarities and transcendence problems from arithmetic dynamics

TL;DR

This paper develops an algebraicity criterion linking the abundance of rational points on analytic subvarieties to their algebraic nature, and applies it to study invariant subvarieties and transcendence in arithmetic dynamics.

Contribution

It introduces a new algebraicity criterion and applies it to classify invariant subvarieties and analyze transcendence problems in polynomial and endomorphism dynamical systems.

Findings

Characterization of algebraic products of Böttcher coordinates and heights.

Partial classification of invariant subvarieties for split polynomial maps.

Results on the algebraicity of periodic curves and their degrees.

Abstract

We introduce an algebraicity criteria. It has the following form: under certain conditions, an analytic subvariety of some algebriac variety over a global field $K$, if it contains many $K$-points, then it is algebraic over $K.$ This gives a way to show the transcendence of points via the transcendence of analytic subvarieties. Such a situation often appears when we have a dynamical system, because we can often produce infinitely many points from one point via iterates. Combing this criteria and the study of invariant subvarieties, we get some results on the transcendence in arithmetic dynamics. We get a characterization of products of B\"ottcher coordinates or products of multiplicative canonical heights for polynomial dynamical pairs to be algebraic. For this, we studied the invariant subvarieties for product of endomorphisms. In particular, we partially generate Medvedev-Scanlon's classification of invariant subvarieties of split polynomial maps to separable endomorphisms on $(\mathbb{P}^1)^N$ in any characteristic. We also get some hight dimensional partial generalization via introducing of a notion of independence. We then study dominant endomorphisms $f$ on $\mathbb{A}^N$ over a number field of algebraic degree $d\geq 2$. We show that in most of the cases (e.g. when such an endomorphism extends to an endomorphism on $\mathbb{P}^N$), there are many analytic curves centered at infinity which are periodic. We show that for most of them, it is algebraic if and only if it contains one algebraic point. We also study the periodic curves. We show that for most of $f$, all periodic curves has degree at most $2$. When $N=2$, we get a more precise classification result. We show that under a condition which is satisfied for a general $f$, if $f$ has infinitely many periodic curves, then $f$ is homogenous up to a changing of origin.