On the construction of the natural extension of the Hurwitz complex continued fraction map
type: Item
instance of: scholarly article
title: On the construction of the natural extension of the Hurwitz complex continued fraction map
zbMATH Open document ID: 1414.11090
author: Hiromi Ei
author: Shunji Ito
author: Hitoshi Nakada
author: Rie Natsui
published in: Monatshefte für Mathematik
publication date: 2019-01-04T00:00:00Z
review text: Let \[ U=\Big\{z \in \mathbb{C} : z=x+iy,-\frac{1}{2}\leq x < \frac{1}{2},\; -\frac{1}{2}\leq y < \frac{1}{2},\Big\} \] and \( [z]_2=[x]_2+i[y]_2 \) for \(z=x+iy,x,y \in \mathbb{R}\), where \([x]_2\) means the nearest integer of \(x\), that is, \([x]_2=k\) when \(x \in [k-\frac{1}{2},k+\frac{1}{2}]\) for \(k \in \mathbb{Z}\). The set of algebraic integers in \(\mathbb{Q}(\sqrt{-1})\) is denoted by \(\mathbb{Z}[i]\). Define the Hurwitz continued fraction map \(T\) by \[ T(z)=\begin{cases} \frac{1}{z}-\Big[ \frac{1}{z}\Big]_2 & \text{if } z\neq 0, \\ 0 & \text{if } z=0 \end{cases} \] for \(z\in U \). We have the Hurwitz continued fraction expansion of \(z \in U \) as \[ z=\frac{\; 1 \; |}{|\, a_1}+\frac{\; 1 \; |}{|\, a_2}+\frac{\; 1 \; |}{|\, a_3}+ \cdots \] with \( a_n=a_n(z)=\left[\frac{1}{T^{n-1}(z)}\right]_2 \; \text{for}\; n \geq 1. \) The expansion terminates in finitely many steps, i.e., \[ z=\frac{\; 1 \; |}{|\, a_1}+\frac{\; 1 \; |}{|\, a_2}+\frac{\; 1 \; |}{|\, a_3}+ \cdots+ \frac{\; 1 \; |}{|\, a_n} \] for some \(n \geq 1\) iff \(z \in \mathbb{Q}(\sqrt{-1}) \cap U\). The existence of an absolutely continuous (with respect to the Lebesgue measure) invariant probability measure for \(T\) was shown in [\textit{H. Nakada}, Keio Eng. Rep. 29, 93--108 (1976; Zbl 0412.10036)] (see also [\textit{H. Nakada}, Sémin. Théor. Nombres, Univ. Bordeaux I 1987--1988, Exp. No. 45, 10 p. (1988; Zbl 0714.11047)]) with its ergodicity. In this article, the density function of the absolutely continuous invariant measure for the map associated to the Hurwitz continued fractions is characterized. A representation of its natural extension map (in the sense of an ergodic measure preserving map) on a subset of \(\mathbb{C}\times \mathbb{C}\) is constructed. This subset is constructed by the closure of pairs of the \(n\)-th iteration of a complex number by the Hurwitz complex continued fraction map and \(-\frac{Q_n}{Q_{n-1}}\), where \(Q_n\) is the denominator of the \(n\)-th converge…
reviewed by: Mykhaylo Pahirya
Mathematics Subject Classification ID: 11K50
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 05B45
Mathematics Subject Classification ID: 52C20
zbMATH DE Number: 6997677
zbMATH Keywords: complex continued fractions
zbMATH Keywords: invariant measure
zbMATH Keywords: natural extension
zbMATH Keywords: tiling
MaRDI profile type: Publication
full work available at URL: s00605-018-1229-0
OpenAlex ID: W2899904828
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Wikidata QID: Q128953192
DOI: 10.1007/S00605-018-1229-0
description: scientific article; zbMATH DE number 6997677