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

Abstract

We consider semigroup dynamical systems defined by several polynomials over a number field 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 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 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).

On Semigroup Orbits of Polynomials and Multiplicative Orders

“…For doing this, we use recent results of Ostafe and Young [4] about the finiteness of cyclotomic algebraic points that are preperiodic for F and that fall on the set of roots of unity, and of graph theory due to Mérai and Shparlinski [2]. Sections 2, 3 and 5 are devoted to preliminar notation and results.…”

Finite orbit points for sets of quadratic polynomials

“…Finally, for those readers interested in additional work on arithmetic aspects of monoid dynamics, see [2,3,10]. We now proceed with the proof of Theorem 1.1.…”

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].…”