Pseudo-Anosov diffeomorphisms in pseudo-surfaces

There are diffeomorphisms on a compact surface S with uniformly hyperbolic 1 dimensional stable and unstable foliations if and only if S is a torus: the Anosov diffeomorphisms. What is happening on the other surfaces? This question leads to the study of pseudo-Anosov maps. Both Anosov and pseudo-Anosov maps appear as periodic points of the geodesic Teichmüller flow T _t on the unitary tangent bundle of the moduli space over S. We observe that the points of pseudo-Anosov maps are regular (the foliations are like the ones for the Anosov automorphisms) except for a finite set of points, called singularities, which are characterized by their number of prongs k. The stable and unstable foliations near the singularities are determined by the real and the imaginary parts of the quadratic differential $\sqrt{z^{k-2}(dz)^{2}}$ . By a coordinate change u(z)=z^k/2 the quadratic differential z^k−2(dz)² gives rise to the quadratic differential (du)² and, in this new coordinates, the pseudo-Anosov maps are uniform contractions and expansions of the stable and unstable foliations. This fact inspired the construction of Pinto-Rand’s pseudo-smooth structures, near the singularities, such that the pseudo-Anosov maps are smooth for this pseudo-smooth structures, and have the property that the stable and unstable foliations are uniformly contracted and expanded by the pseudo-Anosov dynamics. We define a pseudo-linear algebra, the first step in constructing the notion of the derivative of a map at a singularity. In this way, we obtain a pseudo-smooth structure at the singularity, leading to Pinto-Rand’s pseudo-smooth manifolds, pseudo-smooth submanifolds, pseudo-smooth splittings and pseudo-smooth diffeomorphisms. The Stable Manifold Theorem, for pseudo-smooth manifolds, is presented giving the associated pseudo-Anosov diffeomorphisms.