  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.
