Inducing contractions of the mother of all continued fractions

TL;DR

The paper develops contracted Farey expansions by coupling induction on Ito's natural extension of the Farey tent map with contraction of gcf representations, creating a broad, unified class of continued fraction algorithms parameterised by subregions $R$. This framework recovers and interrelates regular continued fractions, $S$-expansions, and Nakada's $\alpha$-continued fractions, while providing a two‑sided shift description for contracted expansions. By translating subsequences of Farey convergents into contracted gcf convergents, the authors connect dynamical properties (ergodicity, entropy) across these families and establish explicit induced natural extensions for each. The results not only unify disparate CF theories but also offer avenues for constructing accelerated expansions with controlled approximation properties and deeper entropy analyses.

Abstract

We introduce a new, large class of continued fraction algorithms producing what are called contracted Farey expansions. These algorithms are defined by coupling two acceleration techniques -- induced transformations and contraction -- in the setting of Shunji Ito's natural extension of the Farey tent map, which generates `slow' continued fraction expansions. In addition to defining new algorithms, we also realise several existing continued fraction algorithms in our unifying setting. In particular, we find regular continued fractions, the second-named author's $S$-expansions, and Nakada's parameterised family of $α$-continued fractions for all $0<α\le 1$ as examples of contracted Farey expansions. Moreover, we give a new description of a planar natural extension for each of the $α$-continued fraction transformations as an explicit induced transformation of Ito's natural extension.

Figures (6)

• Figure 1: Graphs of the Gauss map $G$ (left), Nakada's $\alpha$- cf maps $G_\alpha$ (center), and the Farey tent map $F$ (right).
• Figure 2: Up to null sets, the map $\mathcal{G}$ sends the vertical region $V_k$ to the horizontal region $H_k$.
• Figure 3: From left to right: The sets $V_3\cap H_1,\ \mathcal{F} (V_3\cap H_1),\ \mathcal{F} ^2(V_3\cap H_1)$ and $\mathcal{F} ^3(V_3\cap H_1)$, respectively.
• Figure 4: Left: The domain $\Omega=[0,1]^2$. Right: The first quadrant shows part of the image of $\Omega\backslash ([0,1]\times \{0\})$ under the map $(x,y)\mapsto (x,(1-y)/y)$; the second quadrant shows the image of $\Omega$ under the map $(x,y)\mapsto(x-1,1-y)$.
• Figure 5: Bottom-left: A singularisation area $S$ and its complement $\Delta$ in $\Omega$. Bottom-right: The images of $S$ and $\Delta$ under $\mathcal{G}$. Top-right: The region $R=\psi^{-1}(\Delta)$ and its complement in $H_1$.

Theorems & Definitions (60)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Proposition 4.1
• Remark 5.1
• Lemma 5.2
• proof
• Example 5.3
• Remark 5.4
• Theorem 5.5: Brown--Yin, 1996 BY1996