On the Mozes-Shah phenomenon for horocycle flows on moduli spaces

The Mozes-Shah phenomenon on homogeneous spaces of Lie groups
asserts that the space of ergodic measures under the action by subgroups
generated by unipotents is closed. A key input to their work is Ratner's
fundamental rigidity theorems. Beyond its intrinsic interest, this result
has many applications to counting problems in number theory. The problem
of counting saddle connections on flat surfaces has motivated the search
for analogous phenomena for horocycle flows on moduli spaces of flat
structures. In this setting, Eskin, Mirzakhani, and Mohammadi showed that
this property is enjoyed by the space of ergodic measures under the action
of (the full upper triangular subgroup of) SL(2,R). We will discuss joint
work with Jon Chaika and John Smillie showing that this phenomenon fails
to hold for the horocycle flow on the stratum of genus two flat surfaces
with one cone point. As a corollary, we show that a dense set of horocycle
flow orbits are not generic for any measure; in contrast with Ratner's
genericity theorem.