Multiplicative dependence among iterated values of rational functions modulo finitely generated groups

TL;DR

This paper investigates the multiplicative dependence of iterated values in algebraic dynamical systems over number fields, combining ideas from Northcott's and Siegel's theorems to establish finiteness results.

Contribution

It introduces new results on multiplicative dependence in orbits of rational functions modulo finitely generated groups, blending dynamical systems and Diophantine finiteness theorems.

Findings

Finiteness results for multiplicative dependence in orbits

Connections between dynamical systems and $S$-unit equations

Extensions of classical theorems to algebraic dynamical contexts

Abstract

We study multiplicative dependence between elements in orbits ofalgebraic 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.