Limit Theorems for Horocycle Flows

TL;DR

This paper establishes limit theorems for horocycle flows on compact negatively curved surfaces, using finitely-additive measures and invariant distribution classification to derive asymptotic formulas for ergodic integrals.

Contribution

It introduces a new approach to limit theorems for horocycle flows via finitely-additive measures and builds on the classification of invariant distributions.

Findings

Asymptotic formulas for ergodic integrals derived

Limit theorems established for horocycle flows

Connection to translation flows on flat surfaces

Abstract

The main results of this paper are limit theorems for horocycle flows on compact surfaces of constant negative curvature. One of the main objects of the paper is a special family of horocycle-invariant finitely-additive Hoelder measures on rectifiable arcs. An asymptotic formula for ergodic integrals for horocycle flows is obtained in terms of the finitely-additive measures, and limit theorems follow as a corollary of the asymptotic formula. The objects and results of this paper are similar to those in [15], [16], [4] and [5] for translation flows on flat surfaces. The arguments are based on the classification of invariant distributions for horocycle flows established in [12].