On the construction of the natural extension of the Hurwitz complex continued fraction map
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 convergent of the Hurwitz continued fractions. The absolutely continuous invariant measure for the natural extension map is induced from the invariant measure for Möbius transformations on the set of geodesics over the three-dimensional upper-half space. Then the absolutely continues invariant measure for the Hurwitz continued fraction map is given by its marginal measure. The work contains a lot figures which illustrate the theoretical results.
scientific article; zbMATH DE number 6997677
scientific article