2007/12/08 by John H. Hubbard, Hubbard, John H., Robyn L. Miller +1
Earth and Planetary Sciences · Mathematics · #Aquatic and Environmental Studies #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.0712.1300
openalex publication_date 2007/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 and later reappeared as an illustrative special case of Ratner's work 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. In many aspects of this exposition we are indebted to Bekka and Mayer's more ambitious survey, "Ergodic Theory and Topological Dynamics for Group Actions on Homogeneous Spaces."