THE GEOMETRY OF MEASURED GEODESIC LAMINATIONS AND THE EARTHQUAKE FLOW ON TEICHMUELLER SPACE.

摘要

A measured geodesic lamination (MGL) on a hyperbolic Riemann surface is a closed disjoint union of simple complete geodesics with an invariant transverse measure. A MGL can be quite complicated, for a local cross section typically looks like the union of a Cantor set and a finite set. Thurston showed that the space of all MGLs on a fixed Riemann surface of genus g is homeomorphic to R('6g-6), with the rational points corresponding to simple closed geodesics with Dirac measure.;Kerkhoff showed that the natural map between the space of MGLs on a Riemann surface and the tangent space of Teichmuller space at the surface is a homeomorphism, hence earthquake deformations define a flow on the tangent bundle of Teichmuller space. Kerkhoff also showed that the flow is a real analytic function of time, hence earthquake paths are real analytic.;In my thesis I studied the geometry of MGLs, train tracks and earthquakes. I examined how the leaves of MGLs move about the surface as you change the weights on a train track and showed that if two weights are (epsilon) close then the angle of leaves in the corresponding MGLs are in fact C (epsilon) log('k) (1/(epsilon)) close on the surface. I also showed the following properties of the earthquake flow:;Theorem 1. Earthquake paths are solutions of the second order system of ODEs: (UNFORMATTED TABLE FOLLOWS).;Fenchel and Nielsen studied deformations of Riemann surfaces obtained by cutting a Riemann surface along a simple closed geodesic and gluing the boundary components back with a twist. Thurston generalized the Fenchel-Nielsen deformation to "twisting" along MGL. Such a deformation is called an earthquake.;x(,k) + (GAMMA)(,ij)('k)(x,x)x(,i)x(,j) = 0 k = 1,2,...,n.;(TABLE ENDS).;where the (GAMMA)(,ij)('k) transform as Christoffel symbols but live on the tangent bundle of Teichmuller space.;Theorem 2. The infinitestimal generator of the earthquake flow is not C('2), hence the earthquake flow is not a smooth flow.