Pseudo-Anosov map


In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its definition relies on the notion of a measured foliation introduced by William Thurston, who also coined the term "pseudo-Anosov diffeomorphism" when he proved his classification of diffeomorphisms of a surface.

Definition of a measured foliation

A measured foliation F on a closed surface S is a geometric structure on S which consists of a singular foliation and a measure in the transverse direction. In some neighborhood of a regular point of F, there is a "flow box" φ: U → R2 which sends the leaves of F to the horizontal lines in R2. If two such neighborhoods Ui and Uj overlap then there is a transition function φij defined on φj, with the standard property
which must have the form
for some constant c. This assures that along a simple curve, the variation in y-coordinate, measured locally in every chart, is a geometric quantity and permits the definition of a total variation along a simple closed curve on S. A finite number of singularities of F of the type of "p-pronged saddle", p≥3, are allowed. At such a singular point, the differentiable structure of the surface is modified to make the point into a conical point with the total angle πp. The notion of a diffeomorphism of S is redefined with respect to this modified differentiable structure. With some technical modifications, these definitions extend to the case of a surface with boundary.

Definition of a pseudo-Anosov map

A homeomorphism
of a closed surface S is called pseudo-Anosov if there exists a transverse pair of measured foliations on S, Fs and Fu, and a real number λ > 1 such that the foliations are preserved by f and their transverse measures are multiplied by 1/λ and λ. The number λ is called the stretch factor or dilatation of f.

Significance

Thurston constructed a compactification of the Teichmüller space T of a surface S such that the action induced on T by any diffeomorphism f of S extends to a homeomorphism of the Thurston compactification. The dynamics of this homeomorphism is the simplest when f is a pseudo-Anosov map: in this case, there are two fixed points on the Thurston boundary, one attracting and one repelling, and the homeomorphism behaves similarly to a hyperbolic automorphism of the Poincaré half-plane. A "generic" diffeomorphism of a surface of genus at least two is isotopic to a pseudo-Anosov diffeomorphism.

Generalization

Using the theory of train tracks, the notion of a pseudo-Anosov map has been extended to self-maps of graphs and outer automorphisms of free groups. This leads to an analogue of Thurston classification for the case of automorphisms of free groups, developed by Bestvina and Handel.