Periodic points in complex continued fractions by J. Hurwitz

TL;DR

This paper explores the structure of purely periodic points in complex continued fractions generated by Hurwitz's algorithm, providing explicit descriptions via the natural extension and connecting to ergodic properties.

Contribution

It explicitly characterizes purely periodic points in Hurwitz's complex continued fractions using the natural extension, building on previous ergodic analysis.

Findings

Explicit description of purely periodic points

Connection between periodic points and ergodic properties

Extension of Hurwitz's and Tanaka's work on complex continued fractions

Abstract

J. Hurwitz introduced an algorithm that generates a continued fraction expansion for complex numbers $\alpha \in \mathbb{C}$, where the partial quotients belong to $(1+i)\mathbb{Z}[i]$. J. Hurwitz's work also provides a result analogous to Lagrange's theorem on periodic continued fractions, describing purely periodic points using the dual continued fraction expansion. S. Tanaka \cite{ST} examined the identical algorithm and constructed the natural extension of the transformation generating the continued fraction expansion and established its ergodic properties. In this paper, we aim to describe J. Hurwitz's insights into purely periodic points explicitly through the natural extension.