Train Track Lamination

Transferred from to commons.

Train track lamination.

Via train tracks. Furthermore we show that the weak topology on the measured laminations weakly carried by a train track corresponds to a pointwise weak convergence of the edge weight systems. The weights vary over a convex cone and these cones give coordinate charts for a manifold. The maximum euler characteristic function is continuous on certain parametrized families of lamination links carried by a train track neighborhood.

A lamination of a surface is a partition of a closed subset of the surface into smooth curves. The pil structure on the space of measured geodesic laminations. Given a train track in m satisfying certain nontriviality condi tions then each assignment of positive weights to the various sectors of subject to compatibility conditions at the branching loci deflnes a measured lamination. One application of the dictionary is to the study of thurston s classification of mapping classes on s cb88.

Introduction symbolic dynamical systems were rst introduced more than a century ago to gain insight. S g n s g n homotopic to the identity mapping into and such. The study of train tracks was originally motivated by the following observation. A geodesic lamination on s g n is said to be carried by a train track if there is a di eren tiable map f.

One can also use train track expansions to detect finer properties of a measured lamination such as arationality which means that the lami nation fills the whole surface. We don t know exactly which piece of track will be in these cells but we know they will contain a piece of track so we can mark them as being occupied. If a generic lamination on a surface is looked at from a distance by a myopic person it will look like a train track. Every train track can be turned into a generic train track by sliding its edges.

21 november 2006 original upload date source. David eppstein at english wikipedia. These are cells where there is a piece of track that points to that cell. If each vertical fiber has nontrivial intersection with some leaf then the lamination is fully carried by the train track.

A switch in a train track and the corresponding portion of a lamination. When one considers the teichmüller space t x of the riemann surface x it is natural to restrict the attention to the space ml b x of bounded measured laminations. One of maximum euler characteristic subject to the condition that the seifert lamination is carried by an aspherical branched surface and satisfying fur ther conditions.