Horocycle flow on negative variable curvature surface is standard

TL;DR

This paper offers a simplified proof that the horocycle flow on negatively curved surfaces is standard, illustrating the method's versatility and providing insights into Kakutani equivalence of unipotent flows.

Contribution

It presents a new, simplified proof of the standardness of horocycle flow on variable negative curvature surfaces, extending the method to non-homogeneous flows.

Findings

Simplified proof of horocycle flow standardness

Application of method to non-homogeneous flows

Insights into Kakutani equivalence of unipotent flows

Abstract

We provide a new proof that the horocycle flow preserving the Margulis measure on a variable negative curvature surface is standard. This was first proved by Ratner. The main purpose of this note is to provide a simplified case of the arguments for Kakutani equivalence of unipotent flows on homogeneous spaces, which have similar but more complicated structures, as well as illustrate the versatility of the method by applying it to a non-homogeneous flow.