Non-affine horocycle orbit closures on strata of translation surfaces: new examples

TL;DR

This paper constructs new examples of horocycle-invariant ergodic measures on strata of translation surfaces by integrating specific real vector-fields, revealing novel boundary behaviors of these measures.

Contribution

It introduces a method to generate new horocycle-invariant measures supported on boundary-approaching manifolds in strata of translation surfaces.

Findings

New examples of horocycle-invariant ergodic measures

Measures supported on boundary-approaching manifolds

Novel boundary behavior of measures

Abstract

We investigate specific examples of locally-defined real vector-fields on strata of translation surfaces. Integrating SL(2,R)-loci of Veech surfaces along these vector-fields yield interesting new examples of horocyle-invariant ergodic measures. These measures are supported on closed immersed manifolds with boundary that approach the boundary of strata in a novel way.