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We consider semigroup dynamical systems defined by several polynomials over anumber 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 systemsthat fall in the cyclotomic closure of $\mathbb{K}$. More generally, weconsider the finiteness of initial points in the cyclotomic closure for whichthe orbit contains an algebraic integer of bounded house. This work extendsprevious results for classical obits generated by one polynomial over$\mathbb{K}$ obtained initially by Dvornicich and Zannier (for preperiodicpoints), and then by Chen and Ostafe (for roots of unity and elements ofbounded house in orbits).

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