Flat trace distribution of the geodesic flow on compact hyperbolic plane

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 Poincaré upper half-plane ${\mathbf{H}}^2=\{z\in \mathbb{C}:\mathrm{\Im }(z)>0\}$, equipped with the hyperbolic metric $d{s}^2=\frac{d{z}^2}{\mathrm{\Im }(z{)}^2}$.