Equidistribution of Horocyclic Flows on Complete Hyperbolic Surfaces of Finite Area

TL;DR

This paper explains Ratner's Equidistribution Theorem for horocyclic flows on hyperbolic surfaces, discusses its historical development, and presents a probabilistic result about random walks on the modular surface.

Contribution

It provides a self-contained introduction to a classical equidistribution theorem and includes a new probabilistic insight about random walks on horocycles.

Findings

Horocyclic flows are equidistributed on finite-area hyperbolic surfaces.

Uncentered random walks on horocycles generally fail to equidistribute.

The paper offers an accessible exposition of Ratner's theorem and related results.

Abstract

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."