• geodesic currents
• foliations
• laminations
• horocycle flow
• earthquake
• Teichmüller

The aim of this thesis is to present results proved in a paper by Mirzakhani, which shows that the earthquake flow over Teichmüller space is measurably conjugated to the horocycle flow, which is better understood. In this way earthquake flow also shares measure theoretic properties which were before known only for the horocycle flow. The Teichmüller space T_g of a surface of genus g≥2 is the set of all hyperbolic metrics up to isometries isotopic to the identity. Thurston introduced earthquakes as a class of transformations of the surface’s metric, which in the elementary case are done by twisting the surface along a closed geodesic. In the general case, the twist is performed along a geodesic measured lamination, which is a certain class of geodesically foliated subsets of the surface. We introduce them as boundary of T_g in the Bonahon’s formulation of Thurston compactification theorem of Teichmüller space. Earthquakes induce a Mapping Class Group invariant flow on the bundle of measured laminations on T_g. On the other side, the horocycle flow is an MCG-invariant flow defined on the bundle of holomorphic quadratic differentials QT_g, which is the cotangent bundle of T_g. The proof of the conjugacy make use of previous results by Bonahon and Papadopoulos on symplectic structures of T_g, the space of laminations and the space of transverse cocycles with respect to a lamination which parametrizes both the first two via different versions of shear coordinates.