Rational dynamical systems, $S$-units, and $D$-finite power series

Abstract

Let K be an algebraically closed field of characteristic zero and let G be a finitely generated subgroup of the multiplicative group of K. We consider K-valued sequences of the form a n := f (ϕ n (x 0 )), where ϕ : X X and f : X P 1 are rational maps defined over K and x 0 ∈ X is a point whose forward orbit avoids the indeterminacy loci of ϕ and f . Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the set of n for which a n ∈ G is a finite union of arithmetic progressions along with a set of Banach density zero. In addition, we show that if a n ∈ G for every n and X is irreducible and the ϕ orbit of x is Zariski dense in X then there are a multiplicative torus G d m and maps Ψ :We then obtain results about the coefficients of D-finite power series using these facts. Contents 1. Introduction 1 2. Linear recurrences in abelian groups 5 3. Multiplicative dependence and S-unit equations 11 4. Proof of Theorem 1.1 17 5. Heights of points in orbits 20 6. Applications to D-finite power series 23 References 27