Periodic points, linearizing maps, and the dynamical Mordell–Lang problem

Abstract

Under suitable hypotheses, we prove a dynamical version of the Mordell-Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism Φ : X → X.We also prove a version of the Mordell-Lang conjecture that holds for any endomorphism of a semiabelian variety. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchimedean dynamics.

The algebraic dynamics of generic endomorphisms of ℙn

“…For each x ∈ P n (K), every infinite subset of O f (x), the f -orbit of x, is Zariski dense in P n K . This implies the dynamical "Mordell-Lang" conjecture of Denis [4] and Ghioca-Tucker [9] for generic endomorphisms. This conjecture has been proved for etale endomorphisms of arbitrary varieties by Bell, Ghioca and Tucker [2] but there are only a few other cases where it is known.…”

“…Let (X, f ) be an algebraic dynamical system over a field K of characteristic zero, i.e., X is an algebraic variety and f : X → X is a morphism. The conjecture of Ghioca and Tucker [9] asserts that if x ∈ X(K) and Y a subvariety of X are such that O f (x) ∩ Y (K) is infinite then there is a periodic subvariety Z of X with Z ⊂ Y and Z(K) ∩ O f (x) = ∅. It has been proved when f is etale by Bell, Ghioca and Tucker in [2] and in a few other cases.…”

The orbit intersection problem for linear spaces and semiabelian varieties

“…By [BGT10, Proposition 4.4] (see also [GT09,Proposition 3.3] and [BGT16, Chapter 4]), there exist a prime number p and an embedding of R into Z p such that (i) X has a smooth semiabelian model X over Z p ; (ii) Φ and Ψ extend to endomorphism of X ; (iii) α, α 0 , β, β 0 extend to points in X (Z p ). Let f 0 and g 0 denote the linear maps induced on the tangent space at 0 by Φ 0 , respectively Ψ 0 .…”

“…Motivated by the classical Mordell-Lang conjecture proved by Faltings [Fal94] and Vojta [Voj96], the Dynamical Mordell-Lang Conjecture predicts that for a given x ∈ X(K) and a closed subvariety V of X, the set {n ∈ N : Φ n (x) ∈ V (K)} is a finite union of arithmetic progressions (see [GT09,Conjecture 1.7] along with the earlier work of Denis [Den94] and Bell [Bel06]). Considering X a semiabelian variety and Φ the translation by a point x ∈ X(K), one recovers the cyclic case in the classical Mordell-Lang conjecture from the above stated Dynamical Mordell-Lang Conjecture.…”

Dynamical Mordell-Lang conjecture in positive characteristic

“…Motivated by the classical Mordell-Lang conjecture proved by Faltings [Fal94] (for abelian varieties) and Vojta [Voj96] (for semiabelian varieties), the dynamical Mordell-Lang Conjecture predicts that for a quasiprojective variety X endowed with a self-map Φ defined over a field K of characteristic 0, given a point α ∈ X(K) and a subvariety V of X, the set S := {n ∈ N : Φ n (α) ∈ V (K)} is a finite union of arithmetic progressions (see [GT09,Conjecture 1.7] along with the earlier work of Denis [Den94] and Bell [Bel06]). Considering X a semiabelian variety and Φ the translation by a point x ∈ X(K), one recovers the cyclic case in the classical Mordell-Lang conjecture from the above stated dynamical Mordell-Lang Conjecture; we refer the readers to [BGT16] for a survey of recent work on the dynamical Mordell-Lang conjecture.…”