Fixed points and orbits in skew polynomial rings

TL;DR

This paper extends the study of polynomial dynamics to skew polynomial rings, analyzing fixed points and periodic orbits, and providing conditions for periodicity in this more general algebraic setting.

Contribution

It introduces new results on fixed points and periodic points in skew polynomial rings, generalizing previous work from commutative polynomial rings.

Findings

If f(a) = a, then f^n(a) = a for all n in skew polynomial rings.

Provides a sufficient condition for a point to be r-periodic under a skew polynomial.

Builds on foundational results to extend polynomial dynamics to non-commutative settings.

Abstract

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.