**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
$x=\frac{-1}{a_{1}-\frac{1}{a_{2}-\frac{1}{a_{3}}-...}}$ 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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