Conjugating flows on the moduli of hyperboic and flat surfaces

A measured geodesic lamination on a hyperbolic surface encodes the
horizontal trajectory structure of certain quadratic differentials.
Thurston’s earthquake flow along such a lamination induces a dynamical
system on the moduli space of hyperbolic surfaces sharing many properties
with the classical Teichmüller horocycle flow. Mirzakhani gave a dynamical
correspondence between the earthquake and horocycle flows, defined
Lebesgue-almost everywhere. In this talk, we extend Mirzakhani’s conjugacy
and define an extension of the earthquake flow to an action of the upper
triangular group P in PSL(2,R) mapping certain flow lines to Teichmüller
geodesics. We classify the P-invariant ergodic probability measures as
those coming from affine invariant measures on quadratic differentials and
show that our map is a measurable isomorphism between P actions with
respect to these measures. This is joint work with Aaron Calderon.