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On continued fraction maps associated with a submodule of $\mathfrak o(\sqrt{-3})$

Nakada Hitoshi, Natsui Rie, Toyosumi Mako

TL;DR

This work develops a complex continued fraction theory tied to the submodule $\mathcal J=\eta\cdot\mathfrak o(\sqrt{-3})$ of the Eisenstein field, generalizing Hurwitz’s map to an Eisenstein setting. By constructing a hexagonal fundamental domain $U$ and a nearest-integer map $T$, it obtains $\mathcal J$-expansions for all $z\in\mathbb C$ (away from two boundary points) and proves convergence of the resulting continued fractions, a finite-range structure, and a dual-area representation via a natural extension. It further shows the existence of a finite absolutely continuous invariant measure for $T$, proves ergodicity, and establishes the strict monotonicity of the denominators $|q_n|$ with a positive Lyapunov exponent given by $\lim_{n\to\infty}\frac{1}{n}\log|q_n| = \int_{\hat{U}} \log|w|\, d\hat{\mu}(z,w)$. Collectively, these results provide a robust dynamical framework for complex continued fractions in the Eisenstein setting, with precise geometric and ergodic properties.

Abstract

We define a continued fraction map associated with the $\mathfrak o(\sqrt{-3})$-module $\mathcal J = η\cdot\mathfrak o(\sqrt{-3})$, $η= \frac{3 + \sqrt{-3}}{2}$, which is an Eisenstein field version of the continued fraction map associated with $\mathfrak o(\sqrt{-1}) \cdot (1 + i)$ defined by J.~Hurwitz in the case of the Gaussian field. Together with $T$, we show that all complex numbers $z$ can be expanded as $\mathcal J$-coefficients. We discuss some basic properties of these continued fraction expansions such as the monotonicity of the absolutely value of the principal convergent $q_{n}$ and the existence of the absolutely continuous ergodic invariant probability measure for $T$.

On continued fraction maps associated with a submodule of $\mathfrak o(\sqrt{-3})$

TL;DR

This work develops a complex continued fraction theory tied to the submodule of the Eisenstein field, generalizing Hurwitz’s map to an Eisenstein setting. By constructing a hexagonal fundamental domain and a nearest-integer map , it obtains -expansions for all (away from two boundary points) and proves convergence of the resulting continued fractions, a finite-range structure, and a dual-area representation via a natural extension. It further shows the existence of a finite absolutely continuous invariant measure for , proves ergodicity, and establishes the strict monotonicity of the denominators with a positive Lyapunov exponent given by . Collectively, these results provide a robust dynamical framework for complex continued fractions in the Eisenstein setting, with precise geometric and ergodic properties.

Abstract

We define a continued fraction map associated with the -module , , which is an Eisenstein field version of the continued fraction map associated with defined by J.~Hurwitz in the case of the Gaussian field. Together with , we show that all complex numbers can be expanded as -coefficients. We discuss some basic properties of these continued fraction expansions such as the monotonicity of the absolutely value of the principal convergent and the existence of the absolutely continuous ergodic invariant probability measure for .

Paper Structure

This paper contains 5 sections, 18 theorems, 80 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Convergence of continued fractions
  3. Finite range structure
  4. Dual area
  5. Monotonicity of $|q_{n}|$

Key Result

Theorem 1

For any $z \in U$, there exists a sequence $(b_{n}:n \ge 1)$ such that $b_{n} \in \mathcal{J}$, $n \ge 1$, and for $n\ge 0$ and

Figures (8)

  • Figure 1: $U$ and $U^{-1}= \{z : \tfrac{1}{z} \in U\}$
  • Figure 2: $U_{k, 1}$, $1 \le k \le 5$
  • Figure 3: Finite range structure
  • Figure 4: $V_{0, k, 1}^{*}$, $1 \le k \le 6$ : outside of the gray zone
  • Figure 5: $(V_{0, k, 1}^{*})^{-1}$, $1 \le k \le 6$
  • ...and 3 more figures

Theorems & Definitions (23)

  • Remark 1
  • Theorem 1
  • Lemma 1
  • Remark 2
  • Theorem 2
  • Proposition 1
  • Remark 3
  • Definition 1
  • Lemma 2
  • Lemma 3
  • ...and 13 more