On algebraic integers of bounded house and preperiodicity in polynomial semigroup dynamics

TL;DR

This paper investigates finiteness properties of algebraic integers with bounded house and preperiodic points in polynomial semigroup dynamics over number fields, extending classical results to more complex systems.

Contribution

It extends previous finiteness results from single polynomial dynamics to semigroup systems, analyzing algebraic integers of bounded house within these orbits.

Findings

Finiteness of preperiodic points in the cyclotomic closure.

Finiteness of initial points with orbits containing algebraic integers of bounded house.

Extension of classical results to polynomial semigroup dynamics.

Abstract

We consider semigroup dynamical systems defined by several polynomials over a number field $\mathbb{K}$, and the orbit (tree) they generate at a given point. We obtain finiteness results for the set of preperiodic points of such systems that fall in the cyclotomic closure of $\mathbb{K}$. More generally, we consider the finiteness of initial points in the cyclotomic closure for which the orbit contains an algebraic integer of bounded house. This work extends previous results for classical obits generated by one polynomial over $\mathbb{K}$ obtained initially by Dvornicich and Zannier (for preperiodic points), and then by Chen and Ostafe (for roots of unity and elements of bounded house in orbits).