Weil — Petersson geodesics on the modular flooring
Authors: Vaibhav Gadre
Abstract: We take note of the Weil — Petersson (WP) metric on the modular flooring. We supply WP geodesics to the frequent cowl of the modular flooring and analyse geometric properties of the lifts as paths inside the hyperbolic metric on the frequent cowl. For any pair of distinct components inside the thick part of the frequent cowl, we present that the WP and hyperbolic geodesic segments that be a part of the pair, fellow-travel inside the thick half and all deviations between these segments come up all through cusp excursions. Furthermore, we give a quantitative analysis of the deviation all through an tour. We leverage the man touring to derive a correspondence between recurrent WP and hyperbolic geodesic rays from a base-point. We current that the correspondence is perhaps promoted to a homeomorphism on the circle of directions. By analysing cuspidal winding of a typical WP geodesic ray, we current that the homeomorphism pushes forward a Lebesgue measure on the circle to a singular measure. By means of continued fraction coefficients, the singularity boils proper all the way down to a comparability that we present, particularly, the frequent coefficient is bounded alongside a typical WP ray nonetheless unbounded alongside a typical hyperbolic ray
