Pseudo-Anosov map

Summary

A pseudo-Anosov map is a homeomorphism of a surface that stretches by a factor $\lambda > 1$ along one singular foliation. It contracts by $1/\lambda$ along a transverse one. It generalizes the Anosov diffeomorphisms of the torus to surfaces of higher genus. The price is a finite set of singular points where the foliations have three or more prongs. By the Nielsen–Thurston classification, pseudo-Anosov maps are the typical elements of the mapping class group. They have rich dynamics: positive entropy $\log \lambda$, dense periodic points, and they are the monodromies of hyperbolic 3-manifolds fibering over the circle.