Algebraicity Criteria, Invariant Subvarieties and Transcendence Problems from Arithmetic Dynamics

Abstract

We introduce an algebraicity criterion. It has the following form: Consider an analytic subvariety of some algebraic variety X over a global field K. Under certain conditions, if X contains many K-points, then X 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. Combining this criterion and the study of invariant subvarieties, we get some results on the transcendence in arithmetic dynamics. We get a characterization for products of Böttcher coordinates or products of multiplicative canonical heights for polynomial dynamical pairs to be algebraic. For this, we study the invariant subvarieties for products of endomorphisms. In particular, we partially generalize 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 high dimensional partial generalization via introducing a notion of independence. We then study dominant endomorphisms f on ${\mathbb A}^N$ over a number field of algebraic degree $d\ge 2$. We show that in most 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 at least one algebraic point. We also study the periodic curves. We show that for most f, all periodic curves have 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 change of origin.