Characterising Rigidity and Flexibility of Pseudo-anosov and Other Transversally Laminated Surface Dynamics

We consider surface homeomorphisms f : M ! M which leave invariant a set M and satisfy conditions which generalise the situation where is a hyperbolic invariant set or where f is Anosov or pseudo-Anosov. We deene a C r structure on which gives a notion of smoothness for pseudo-Anosov diieomorphisms with singularities. We prove that the holonomy maps are C 1+ for such systems. We show that there is a one-to-one correspondences between C 1+-conjugacy classes of f and various classes of C 1+ structures associated with , the pair of stable and unstable laminations and the surface M. These include C 1+ structures on in which f is smooth, hyperbolic C 1+ foliated structures, hyperbolic C 1+ transversal structures, hyperbolic C 1+ ho-lonomy structures, hyperbolic HR-structures, hyperbolic HA-structures and HH older solenoid function pairs. We show that all of these are in one-to-one correspondence with C 1+ conjugacy classes of pairs of Markov maps and interval exchange maps on certain train-tracks that we construct. The various structures here are introduced because while some are intrinsic to , others are intrinsic to the pair of laminations or the surface M. This allows us to make the transition between one and two dimensional dynamics. The HH older solenoid function pairs provide a natural moduli space for C 1+ conjugacy classes. A natural completion of this moduli space is given by the C u structures that we introduce (interpreted as the set of hyperbolic C 1+ transversal structures) Using the aane structure on leaves given by the HR-structures we discuss how to construct a canonical orthogonal atlas in which the smoothness of the holonomy is maximised. We discuss related results to those above for C r. We prove rigidity results for those f whose system of holonomy maps have derivative with a modulus of continuity given by the Hausdorr dimensions of the stable and unstable leaves in. We prove that if such a map f is C r and has an aane topological model fa then f is rigid in the sense that it is C 1+-conjugate to fa. If = M then fa is unique in its topological class, up to aane conjugacy. If 6 = M then, up to aane conjugacy, the set of aane maps fa topologically conjugate to f is nite-dimensional. We discuss the relation of these rigidy results and the moduli spaces and discuss the nature …