Inmathematics, specifically intopology, apseudo-Anosov map is a type of adiffeomorphism orhomeomorphism of asurface. It is a generalization of a linearAnosov diffeomorphism of thetorus. Its definition relies on the notion of ameasured foliation introduced byWilliam Thurston, who also coined the term "pseudo-Anosov diffeomorphism" when he proved hisclassification of diffeomorphisms of a surface.

Ameasured foliationF on a closed surfaceS is a geometric structure onS which consists of a singularfoliation and a measure in the transverse direction. In some neighborhood of a regular point ofF, there is a "flow box"φ:U →R² which sends the leaves ofF to the horizontal lines inR². If two such neighborhoodsU_i andU_j overlap then there is atransition functionφ_ij defined onφ_j(U_j), with the standard property

${\displaystyle {\varphi }_{ij}\circ {\varphi }_j={\varphi }_i,}$

which must have the form

${\displaystyle \varphi (x,y)=(f(x,y),c\pm y)}$

for some constantc. This assures that along a simple curve, the variation iny-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 onS. A finite number of singularities ofF 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 angleπp. The notion of a diffeomorphism ofS is redefined with respect to this modified differentiable structure. With some technical modifications, these definitions extend to the case of a surface with boundary.

A homeomorphism

${\displaystyle f:S\to S}$

of a closed surfaceS is calledpseudo-Anosov if there exists a transverse pair of measured foliations onS,F^s (stable) andF^u (unstable), and a real numberλ > 1 such that the foliations are preserved byf and their transverse measures are multiplied by 1/λ andλ. The numberλ is called thestretch factor ordilatation off.

Thurston constructed a compactification of theTeichmüller spaceT(S) of a surfaceS such that the action induced onT(S) by any diffeomorphismf ofS extends to a homeomorphism of the Thurston compactification. The dynamics of this homeomorphism is the simplest whenf 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 thePoincaré half-plane. A "generic" diffeomorphism of a surface of genus at least two is isotopic to a pseudo-Anosov diffeomorphism.

Using the theory oftrain tracks, the notion of a pseudo-Anosov map has been extended to self-maps of graphs (on the topological side) and outer automorphisms offree groups (on the algebraic side). This leads to an analogue of Thurston classification for the case of automorphisms of free groups, developed byBestvina and Handel.

• A. Casson, S. Bleiler, "Automorphisms of Surfaces after Nielsen and Thurston", (London Mathematical Society Student Texts 9), (1988).
• A. Fathi, F. Laudenbach, andV. Poénaru, "Travaux de Thurston sur les surfaces," Asterisque, Vols. 66 and 67 (1979).
• R. C. Penner. "A construction of pseudo-Anosov homeomorphisms", Trans. Amer. Math. Soc., 310 (1988) No 1, 179–197
• Thurston, William P. (1988), "On the geometry and dynamics of diffeomorphisms of surfaces",Bulletin of the American Mathematical Society, New Series,19 (2):417–431,doi:10.1090/S0273-0979-1988-15685-6,ISSN 0002-9904,MR 0956596

• Dynamical systems
• Geometric topology
• Homeomorphisms

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