Fixed points and orbits in skew polynomial rings

In this paper, 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 fon(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.