The Collatz map analogue in polynomial rings and in completions

Abstract

We study an analogue of the Collatz map in the polynomial ring $R[x]$, where $R$ is an arbitrary commutative ring. We prove that if $R$ is of positive characteristic, then every polynomial in $R[x]$ is eventually periodic with respect to this map. This extends previous works of the authors and of Hicks, Mullen, Yucas and Zavislak, who studied the Collatz map on $\mathbb{F}_p[x]$ and $\mathbb{F}_2[x]$, respectively. We also consider the Collatz map on the ring of formal power series $R[[x]]$ when $R$ is finite: we characterize the eventually periodic series in this ring, and give formulas for the number of cycles induced by the Collatz map, of any given length. We provide similar formulas for the original Collatz map defined on the ring $\mathbb{Z}_2$ of $2$-adic integers, extending previous results of Lagarias.