Horocycle flow orbits and lattice surface characterizations

TL;DR

This paper explores the behavior of horocycle flow orbits on translation surfaces, revealing that their closures match $SL_2( eal)$ orbit closures and providing new ways to characterize lattice surfaces.

Contribution

It establishes that horocycle flow orbit closures coincide with $SL_2( eal)$ orbit closures for translation surfaces, offering novel characterizations of lattice surfaces.

Findings

Horocycle flow orbit closures equal $SL_2( eal)$ orbit closures.

New characterizations of lattice surfaces based on horocycle flow.

Results hold in almost all directions.

Abstract

The orbit closure of any translation surface under the horocycle flow in almost any direction equals its $SL_2(\mathbb{R})$ orbit closure. This result gives rise to new characterizations of lattice surfaces in terms of the hororcycle flow.