Fixed points and orbits in skew polynomial rings

Abstract

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring [Formula: see text]. We extend results of the first author and Vishkautsan on polynomial dynamics in [Formula: see text]. In particular, we show that if [Formula: see text] and [Formula: see text] satisfy [Formula: see text], then [Formula: see text] for every formal power of [Formula: see text]. More generally, we give a sufficient condition for a point [Formula: see text] to be [Formula: see text]-periodic with respect to a polynomial [Formula: see text]. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy.