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

TL;DR

This paper proves an extreme value law for geodesic excursions on a hyperbolic surface related to the theta group, using a novel continued fraction algorithm and transfer operator spectral analysis.

Contribution

It introduces a generalized continued fraction system combining even and odd-odd maps, linking it to geodesic flow and deriving a new extreme value law for cusp excursions.

Findings

Established an extreme value law for geodesic excursions.

Linked continued fraction digits to cusp excursions.

Demonstrated spectral properties of the transfer operator.

Abstract

We establish an extreme value theorem for the geodesic flow on the hyperbolic surface $\Theta\backslash\mathbb{H}^2$ associated with the theta group $\Theta$. 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 $\Theta\backslash\mathbb{H}^2$.