Hecke groups, linear recurrences, and Kepler limits

We study the linear fractional transformations in the Hecke group $G(\mathrm{\Phi })$ where $\mathrm{\Phi }$ is either root of $x^2-x-1$ (the larger root being the “golden ratio” $\varphi =2\cos \frac{\pi }{5}$.) Let $g\in G(\mathrm{\Phi })$ and let $z$ be a generic element of the upper half-plane. Exploiting the fact that $\Phi^2 = \Phi -1$,wefindthat$g(z)$isaquotientoflinearpolynomialsin$z$ such that the coefficients of $z^1$ and $z^0$ in the numerator and denominator of $g(z)$ appear themselves to be linear polynomials in $\mathrm{\Phi }$ with coefficients that are certain multiples of Fibonacci numbers. We make somewhat less detailed observations along similar lines about the functions in $G(2 \cos \frac {\pi}k)$for$k \geq 5$.