Pacific Dynamics Seminar: James Farre

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.