  We study orbits and fixed points of polynomials in a general skew polynomial
ring $D[x,\sigma, \delta]$. We extend results of the first author and
Vishkautsan on polynomial dynamics in $D[x]$. In particular, we show that if $a
\in D$ and $f \in D[x,\sigma,\delta]$ satisfy $f(a) = a$, then $f^{\circ n}(a)
= a$ 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.
