Fixed points and orbits in skew polynomial rings

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.