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| 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''' invented by [[William Thurston]], who also coined the term "pseudo-Anosov diffeomorphism" when he proved his [[Nielsen–Thurston classification|classification of diffeomorphisms of a surface]].
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| == Definition of a measured foliation ==
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| 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'' → '''R'''<sup>2</sup> which sends the leaves of ''F'' to the horizontal lines in '''R'''<sup>2</sup>. If two such neighborhoods ''U''<sub>''i''</sub> and ''U''<sub>''j''</sub> overlap then there is a '''transition function''' ''φ''<sub>''ij''</sub> defined on ''φ''<sub>''j''</sub>(''U''<sub>''j''</sub>), with the standard property
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| : <math> \phi_{ij}\circ\phi_j=\phi_i,</math> | |
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| which must have the form
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| : <math> \phi(x,y)=(f(x,y),c\pm y)</math> | |
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| 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. 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.
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| == Definition of a pseudo-Anosov map ==
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| A homeomorphism
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| :<math>f: S \to S </math> | |
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| of a closed surface ''S'' is called '''pseudo-Anosov''' if there exists a transverse pair of measured foliations on ''S'', ''F''<sup>''s''</sup> (stable) and ''F''<sup>''u''</sup> (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''. The number ''λ'' is called the '''stretch factor''' or '''dilatation''' of ''f''.
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| == Significance ==
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| Thurston constructed a compactification of the [[Teichmüller space]] ''T''(''S'') of a surface ''S'' such that the action induced on ''T''(''S'') 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.
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| == Generalization ==
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| Using the theory of '''train tracks''', the notion of a pseudo-Anosov map has been extended to self-maps of graphs (on the topological side) and outer automorphisms of [[free group]]s (on the algebraic side). This leads to an analogue of Thurston classification for the case of automorphisms of free groups, developed by [[Mladen Bestvina|Bestvina]] and Handel.
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| ==References==
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| *A. Casson, S. Bleiler, "Automorphisms of Surfaces after Nielsen and Thurston", (London Mathematical Society Student Texts 9), (1988).
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| *A. Fathi, F. Laudenbach, and [[Valentin Poénaru|V. Poénaru]], "Travaux de Thurston sur les surfaces," Asterisque, Vols. 66 and 67 (1979).
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| *R.C. Penner. "A construction of pseudo-Anosov homeomorphisms", Trans. Amer. Math. Soc., 310 (1988) No 1, 179–197
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| *{{Citation | last1=Thurston | first1=William P. | author1-link=William Thurston | title=On the geometry and dynamics of diffeomorphisms of surfaces | doi=10.1090/S0273-0979-1988-15685-6 | mr=956596 | year=1988 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=19 | issue=2 | pages=417–431}}
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| {{DEFAULTSORT:Pseudo-Anosov Map}}
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| [[Category:Dynamical systems]]
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| [[Category:Geometric topology]]
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| [[Category:Homeomorphisms]]
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