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 $f_q:I\lambda \to I_\lambda$, where $\lambda=\lambda_{q=2\cos\frac{\pi}{q},} \,q=3,4,..$ and $I_{\lambda=-\frac{\lambda}{2},\frac{\lambda}{2}$} with $f_q(x)=-\frac{1}{x}-\lfloor-\frac{1}{x\lambda} + \frac{1}{2}\rfloor \lambda$. This map generates the Hurwitz-Nakada continued fraction expansion $x=\frac{-1}{a₁-\frac{1}{a₂-\frac{1}{a_3}-...}}$ with entries $a_i \in \mathbb{Z}\setminus\{0\}$. 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 $G_q =\langle T_{\lambda,S\rangle$.} We furthermore show that the natural extension $F_{q:\Omega\to\Omega$} of the interval map $f_q:I\lambda \to I_\lambda$, 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 $G_{q\setminus\mathbb{H}$.}

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