Fixed points and orbits in skew polynomial rings

Abstract

We study orbits and fixed points of polynomials in a general skew polynomial ring D[x, σ, δ]. We extend results of the first author and Vishkautsan on polynomial dynamics in D[x]. In particular, we show that if a ∈ D and f ∈ D[x, σ, δ] satisfy f (a) = a, then f •n (a) = a for every formal power of f . More generally, we give a sufficient condition for a point a to be r-periodic with respect to a polynomial f . Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy.