Dynamical Systems and Number Theory

# The Hurwitz-Nakada continued fraction and the geodesic flow on Hecke surfaces

speaker: Dieter Mayer (Institute for Theoretical Physics, University of Clausthal)

abstract: We discuss some properties of the map $fq:I\lambda \to I\lambda$, where $\lambda=\lambdaq=2\cos\frac{\pi}{q}, \,q=3,4,..$ and $I\lambda=-\frac{\lambda}{2},\frac{\lambda}{2}$ with $fq(x)=-\frac{1}{x}-\lfloor-\frac{1}{x\lambda} + \frac{1}{2}\rfloor \lambda$. This map generates the Hurwitz-Nakada continued fraction expansion $x=\frac{-1}{a1-\frac{1}{a2-\frac{1}{a3}-...}}$ with entries $ai \in \mathbb{Z}\setminus\{0\}$. We generalize to general $q$ a result of Hurwitz for $q=3$ which says that there is exactly one pair of numbers in $I\lambda$ with purely periodic continued fraction expansion which are equivalent with respect to the Hecke triangle group $Gq =\langle T\lambda,S\rangle$. We furthermore show that the natural extension $Fq:\Omega\to\Omega$ of the interval map $fq:I\lambda \to I\lambda$, where $\Omega$ is some closed subset of $\mathbb{R}2$, can be identified as the Poincare map for the geodesic flow on the Hecke surface $Gq\setminus\mathbb{H}$.

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