Flat trace distribution of the geodesic flow on compact hyperbolic plane

TL;DR

This paper analyzes the spectral properties and flat-trace distribution of the geodesic flow on a compact hyperbolic surface, providing a detailed spectral decomposition of the associated Koopman operator.

Contribution

It establishes the spectral decomposition of the Koopman operator and characterizes the flat-trace distribution for the geodesic flow on a compact hyperbolic surface.

Findings

Spectral decomposition of the Koopman operator for the geodesic flow.

Determination of the flat-trace distribution in this setting.

Insights into the spectral structure of hyperbolic geodesic flows.

Abstract

In this paper, we establish the spectral decomposition of the Koopman operator and determine the flat-trace distribution associated with the geodesic flow on the co-circle bundle over the compactification of Poincar\'e upper half-plane $\mathbf{H}^2 = \{z \in \mathbb{C} : \Im(z) > 0\}$, equipped with the hyperbolic metric $ds^2 = \frac{dz^2}{\Im(z)^2}$.