Dynamics of geodesic and horocyclic flows (Q1642183)

This paper was prepared from notes for a conference talk and the results are presented informally. The author's intention is to describe a unified picture of the ergodic and dynamical properties of the horocyclic flow. The aim is to show the deep connections between the ergodic properties of the horocyclic flow and those of the geodesic flow. The geodesic and horocyclic flows on hyperbolic surfaces are classical examples of flows on homogeneous spaces. Their dynamical and ergodic properties were first explored by G. Hedlund and Hopf in the 1930s. A compact hyperbolic surface \(S\) is the quotient of the Poincaré upper-half plane by a cocompact torsion-free Fuchsian group \(\Gamma\). Its unit tangent bundle \(T'S\), which can be seen as the quotient of PSL\((2,\mathbb{R})\) by \(\Gamma\), is the phase space of these flows. The geodesic flow on \(T'S\) is uniformly hyperbolic with horocyclic orbits as the stable manifolds. The author notes that the hyperbolicity of the geodesic flow leads to infinitely many periodic orbits and ergodic invariant probability measures. The horocyclic flow, by contrast, has very few invariant measures. The primary parts of the paper describe the topological dynamics of the horocyclic flow and the invariant measures for the horocyclic flow with particular attention to the Liouville measure. Along the way the author provides an extended summary of prior results in this area. For the entire collection see [Zbl 1395.37001].