Horocycle dynamics in rank one invariant subvarieties I: weak measure classification and equidistribution

TL;DR

This paper investigates the dynamics of horocycle flows on invariant subvarieties of the moduli space of translation surfaces, establishing measure classification, equidistribution, and counting results that extend previous work.

Contribution

It introduces a weak classification of horocycle invariant measures and applies it to rank one invariant subvarieties, extending prior dynamical results.

Findings

Proves genericity of horocycle orbits and equidistribution.

Establishes asymptotic equidistribution of pushed measures.

Provides counting results for saddle connection holonomies.

Abstract

Let M be an invariant subvariety in the moduli space of translation surfaces. We contribute to the study of the dynamical properties of the horocycle flow on M. In the context of dynamics on the moduli space of translation surfaces, we introduce the notion of a 'weak classification of horocycle invariant measures' and we study its consequences. Among them, we prove genericity of orbits and related uniform equidistribution results, asymptotic equidistribution of sequences of pushed measures, and counting of saddle connection holonomies. As an example, we show that invariant varieties of rank one, Rel-dimension one and related spaces obtained by adding marked points satisfy the 'weak classification of horocycle invariant measures'. Our results extend prior results obtained by Eskin-Masur-Schmoll, Eskin-Marklof-Morris, and Bainbridge-Smillie-Weiss.