Geodesic laminations as geometric realizations of Pisot substitutions

We construct a geodesic lamination on the hyperbolic disk and a dynamical system defined on this lamination. We prove that this dynamical system is a geometrical realization of the symbolic dynamical system that arises from the following Pisot substitution: $1\rightarrow 12, \dotsc, (n-1) \rightarrow 1n, n\rightarrow 1$.

Bending in the space of quasi-Fuchsian structures

In [2] I described the deformation of bending a hyperbolic manifold along an embedded totally geodesic hypersurface. As I remarked there, the deformation is particularly interesting in the case of a surface, because a surface contains many embedded totally geodesic hypersurfaces, namely simple closed curves, along which it is possible to bend. Furthermore, for a surface it is possible to extend the definition of bending to the case of a geodesic lamination, by using the fact that the set of simple closed geodesies is dense in the space of geodesic laminations. This direction has been developed by Epstein and Marden in [1].

Non-smooth geodesic flows and the earthquake flow on Teichmüller space

Thurston generalized the notion of a twist deformation about a simple closed geodesic on a hyperbolic Riemann surface to a twisting or shearing along a much more complicated object called a measure geodesic lamination. This new deformation is called an earthquake and it generates a flow on the tangent bundle of Teichmüller space.In this paper we study the earthquake flow. We show that the flow is not smooth and that it is not the geodesic flow for an affine connection. We also derive the explicit form of the system of differential equations which earthquake trajectories satisfy.