2022/05/05 by Jason P. Bell, Dragos Ghioca, Bell, Jason +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2205.02644
openalex publication_date 2022/05/05 · openalex created_date 2022/05/08 · openalex updated_date 2026/07/28
We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let K be an algebraically closed field of positive characteristic, let G be a finitely generated subgroup of the multiplicative group of K, and let X be a (irreducible) quasiprojective variety defined over K. We consider K-valued sequences of the form an:=f(φn(x0)), where φ\colon X→ X and f\colon X→ℙ1 are rational maps defined over K and x0∈ X is a point whose forward orbit avoids the indeterminacy loci of φ and f. We show that the set of n for which an∈ G is a finite union of arithmetic progressions along with a set of upper Banach density zero. In addition, we show that if an∈ G for every n and the φ orbit of x is Zariski dense in X then there is a multiplicative torus \mathbbGmd and maps Ψ:\mathbbGmd → \mathbbGmd and g:\mathbbGmd → \mathbbGm such that an = g∘ Ψn(y) for some y∈ \mathbbGmd. We then describe various applications of our results.