The arithmetic of semigroup dynamical systems over the cyclotomic closure of a number field

Abstract:

A famous result of Northcott implies that a polynomial defined over a number field KK has only finitely many preperiodic points lying in KK. It is of interest to determine if similar finiteness results hold over certain infinite extensions. Using tools from arithmetic geometry, Dvornicich and Zannier established finiteness of preperiodic points over the cyclotomic closure of a number field. We discuss some of the background of this work, and extend it to the more general dynamical setting of a semigroup of polynomials, generated under composition. More generally, we consider the finiteness of initial points in the cyclotomic closure for which such a system contains an algebraic integer of bounded house. This extends results for classical dynamical systems obtained by Ostafe (in the special case of roots of unity) and Chen.