Fixed points and orbits in skew polynomial rings

Adam Chapman; Elad Paran

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arXiv · 2211.08100

Fixed points and orbits in skew polynomial rings

Adam Chapman·Elad Paran

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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.

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BibTeX RIS

Adam Chapman, Elad Paran. 2022-11-15. Fixed points and orbits in skew polynomial rings. https://arxiv.org/abs/2211.08100

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