Equidistribution of Horocyclic Flows on Complete Hyperbolic Surfaces of Finite Area

We provide a self-contained, accessible introduction to Ratner’s Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. This equidistribution result was first obtained in the early 1980s by Dani and Smillie [DS84] and later reappeared as an illustrative special case [Rat92] of Ratner’s work [Rat91-Rat94] on the equidistribution of unipotent flows in homogeneous spaces. We also prove an interesting probabilistic result due to Breuillard: on the modular surface an arbitrary uncentered random walk on the horocycle through almost any point will fail to equidistribute, even though the horocycles are themselves equidistributed [Bre05]. In many aspects of this exposition we are indebted to Bekka and Mayer’s more ambitious survey [BM00], Ergodic Theory and Topological Dynamics for Group Actions on Homogeneous Spaces. 1 Horocycle flow on hyperbolic surfaces Let X be a complete hyperbolic surface, perhaps the hyperbolic plane H , and let X denote the unit tangent bundle T (X) to X (and H = T H). There are three flows on X which will concern us here. They are realized by three cars, as represented in Figure 1.