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Geodesic flows and the mother of all continued fractions

Claire Merriman

Abstract

We extend the Series' connection between the modular surface $\mathcal{M}=\operatorname{PSL}(2,\mathbb{Z})\backslash\mathbb{H}$, cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions with digits $(1,+1)$ and $(2,-1)$ in the notation used for the Lehner continued fractions. We also introduce an alternative insertion and singularization algorithm for Farey expansions and other non-semiregular continued fractions, and an alternative dual expansion to the Farey expansions so that $\frac{dxdy}{(1+xy)^2}$ is invariant under the natural extension map.

Geodesic flows and the mother of all continued fractions

Abstract

We extend the Series' connection between the modular surface , cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions with digits and in the notation used for the Lehner continued fractions. We also introduce an alternative insertion and singularization algorithm for Farey expansions and other non-semiregular continued fractions, and an alternative dual expansion to the Farey expansions so that is invariant under the natural extension map.

Paper Structure

This paper contains 12 sections, 18 theorems, 53 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Lehner expansions
  3. Farey continued fractions
  4. Cutting Sequences and $\mathcal{M}$
  5. The group $\operatorname{PSL}(2,\mathbb Z)$ and $\mathcal{M} =\operatorname{PSL}(2,\mathbb Z)\backslash\mathbb{H}$
  6. Cutting sequences and continued fraction expansions
  7. Connection with cutting sequence and regular continued fractions
  8. Lehner continued fractions
  9. Farey continued fractions
  10. Applications
  11. An alternate dual expansion to the Farey expansions
  12. Acknowledgements

Key Result

Theorem \oldthetheorem

[Section cut] Our classification of geodesics $\bar{\gamma}$ on $T_1\mathcal{M}$ with cutting sequence $\dots L^{n_{-1}}R^{n_0}L^{n_1}\dots$ depends on whether or not $n_0=1$. In the one line notation given in lehnerdef and fareydef, if $n_0=1$, $\bar{\gamma}$ has a lift on $\mathbb{H}$ with forward and backwards endpoint when $n_{-1}\geq 2$ and when $n_{-1}=1$. When $n_0>1$, $\bar{\gamma}$ has

Figures (5)

  • Figure 1: The edges of the ideal triangles are images of $i\mathbb R$ under the $\operatorname{PSL}(2,\mathbb Z)$ action on $\mathbb{H}$. The edges connect two rational numbers if and only if they are adjacent in some Farey sequence $F_n=\{\frac{p}{q}:0\leq q\leq n\}$. Here is the Farey tessellation up to $n=3$.
  • Figure 2: The fundamental domain ${\mathfrak F}$ is shown in grey in the image on the left. The fundamental Farey cell $\Delta$ is shown in grey on the right.
  • Figure 3: $\gamma$ (solid) has cutting sequence $\dots L R L^2 R \xi_\gamma L \eta_\gamma L\dots$, $\bar{\rho}(\gamma)$ (dashed) has cutting sequence $\dots L R L^2 R \xi_{\bar{\rho}(\gamma)} R \eta_{\bar{\rho}(\gamma)} R^2\dots$
  • Figure 4: $\gamma$ (solid) has cutting sequence$\dots L R L^2 R \xi_\gamma R \eta_\gamma R^2\dots$, $\bar{\rho}(\gamma)$ (dashed) has cutting sequence $\dots LR L \xi_{\bar{\rho}(\gamma)} L \eta_{\bar{\rho}(\gamma)L }\dots$
  • Figure 5: Geodesic with cutting sequence $\dots L^2 R\xi_\gamma R\eta_\gamma R\dots$

Theorems & Definitions (30)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • ...and 20 more