Extreme value theorem for geodesic flow on the quotient of the theta group

Abstract

We establish an extreme value theorem for the geodesic flow on the hyperbolic surface $Θ\backslash\mathbb{H}^2$ associated with the theta group $Θ$. To capture excursions into both cusps of this surface, we introduce a generalized continued fraction algorithm obtained by splicing the even and odd-odd continued fraction maps into a single dynamical system. We prove that the natural extension of this map is isomorphic to the first return map of the geodesic flow on a suitable cross section. Using spectral properties of the associated transfer operator, we derive a Galambos-type extreme value law for the digits of the spliced continued fraction. This symbolic result is then translated into a geometric extreme value theorem describing maximal cusp excursions of geodesics on $Θ\backslash\mathbb{H}^2$.