Finite Partitions for Several Complex Continued Fraction Algorithms

Abstract

We present a property satisfied by a large variety of complex continued fraction algorithms (the “finite building property”) and use it to explore the structure of bijectivity domains for natural extensions of Gauss maps. Specifically, we show that these domains can each be given as a finite union of Cartesian products in $\mathbb{C}\times \mathbb{C}.$ In one complex coordinate, the sets come from explicit manipulation of the continued fraction algorithm, while in the other coordinate the sets are determined by experimental means.

Keywords:

• Continued fractions
• Gauss map
• complex continued fractions
• natural extension
• partition

Notes

1 Dani and Nogueira denote a choice function by f(x). Katok and Ugarcovici use the notation ${\lfloor x\rceil }_{a,b}$ for their “generalized integer part” function.

2 Using a different value of a would give a different ${\mathcal{P}}_1,$ but in the end some ${\mathcal{P}}_n$ would still be exactly the partition from (4–1(4–1) $$\begin{array}{cc}{K}_1\hfill & =\left\{z\in K:\text{Re}z\le 0,\left|z-\mathbb{i}\right|\ge 1,\left|z+\mathbb{i}\right|\ge 1\right\}\hfill \\ K_2\hfill & =\left\{z\in K:\left|z-\mathbb{i}\right|\le 1,\left|z+1\right|\le 1,\left|z-(-1+\mathbb{i})\right|\ge 1\right\}\hfill \\ K_3\hfill & =\left\{z\in K:\left|z-(-1+\mathbb{i})\right|\le 1\right\}\hfill \\ K_i\hfill & =-\mathbb{i}{K}_{i-3}\hspace{1em}\text{for}i=4,\dots ,12.\hfill \end{array}$$(4–1) ).

3 In [14], $G_{a,b}$ is denoted ${\widehat{f}}_{a,b},$ and the set ${\Omega }_{a,b}$ is $\left\{(x,-1/y):(x,y)\in {\widehat{\Lambda }}_{a,b}\right\}.$

4 Brothers Adolf and Julius Hurwitz both studied continued fractions. The term “Hurwitz algorithm” generally refers to the nearest integer algorithm, while the nearest even algorithm (Section 4.2) is sometimes called the “J. Hurwitz algorithm” [14].