In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface.
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(Uj), 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 (i.e. independent of the chart) 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.
A homeomorphism of a closed surface S is called pseudo-Anosov if there exists a transverse pair of measured foliations on S, Fs (stable) and Fu (unstable), 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.
This leads to an analogue of Thurston classification for the case of automorphisms of free groups, developed by Bestvina and Handel.