Multiplicative Dependence Among Iterated Values of Rational Functions Modulo Finitely Generated Groups

Abstract

We study multiplicative dependence between elements in orbits of algebraic dynamical systems over number fields modulo a finitely generated multiplicative subgroup of the field. We obtain a series of results, many of which may be viewed as a blend of Northcott's theorem on boundedness of preperiodic points and Siegel's theorem on finiteness of solutions to S-unit equations.

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

“…From this point of view, it is then natural to consider when the dynamical sequences we study take values in a finitely generated multiplicative group. This question and related questions have already been considered in the case of self-maps of P 1 in [BOSS,OSSZ19].…”

Cyclotomic preperiodic points for morphisms in affine spaces and preperiodic points with bounded house and height

“…Lastly, Section 4 is also place for extending [1,Theorem 1.10]. This result says that if f ∈ K(X) has degree at least 2 and F (T 1 , ..., T k ) = r i=1 c i j∈J i F j with J 1 ∪ ... ∪ J r = {1, ..., k} a disjoint partition and c 1 , ..., c k ∈ K * , then the set of α ∈ K such that there exist positive integers n 1 < n 2 ... < n k satisfying F (f (n 1 ) (α), ..., f (n k ) )(α)) = 0 is a set of bounded height, and there are only finitely many possible values of n 1 , ..., n k that satisfy such equation.…”

“…We extend this result for A N (K) as a ring and F a more general polynomial with split variables. For this, we use the ideas of [1] with properties from the canonical heights of [7,8]. We finish Section 4 with simple semigroup extensions of some results of [9] that have some flavour of Section 3, now also related to Lattès maps.…”

Unlikely intersections over finite fields: Polynomial orbits in small subgroups

“…Recently, several of so called unlikely intersection type results, see [28] for a general background, have been obtained on the scarcity of elements in orbits of polynomial maps in fields of characteristic zero that fall in a set of prescribed additive, multiplicative or algebraic structure. Examples of such sets include ‚ algebraic varieties [1,18,24,26,27] where the problem is also known as the dynamical Mordell-Lang conjecture; ‚ an orbit generated by another polynomial or rational function [9,10]; ‚ the set of all roots of unity in C, see [8,15] and more generally, of algebraic numbers with all conjugates bounded by some constant, see [6,20]; ‚ the set of all perfect powers in a number field, see [3,17]; ‚ a finitely generated group in a number field, see [2,11,19].…”