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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 edit

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: UR2 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. 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 edit

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. The number λ is called the stretch factor or dilatation of f.

Significance edit

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.

Generalization edit

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 groups (on the algebraic side). This leads to an analogue of Thurston classification for the case of automorphisms of free groups, developed by Bestvina and Handel.

References edit

  • A. Casson, S. Bleiler, "Automorphisms of Surfaces after Nielsen and Thurston", (London Mathematical Society Student Texts 9), (1988).
  • A. Fathi, F. Laudenbach, and V. 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

pseudo, anosov, mathematics, specifically, topology, pseudo, anosov, type, diffeomorphism, homeomorphism, surface, generalization, linear, anosov, diffeomorphism, torus, definition, relies, notion, measured, foliation, introduced, william, thurston, also, coin. 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 Contents 1 Definition of a measured foliation 2 Definition of a pseudo Anosov map 3 Significance 4 Generalization 5 ReferencesDefinition of a measured foliation editA 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 f 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 fij defined on fj Uj with the standard property ϕij ϕj ϕi displaystyle phi ij circ phi j phi i nbsp which must have the form ϕ x y f x y c y displaystyle phi x y f x y c pm y nbsp 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 pp 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 editA homeomorphism f S S displaystyle f S to S nbsp 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 l gt 1 such that the foliations are preserved by f and their transverse measures are multiplied by 1 l and l The number l is called the stretch factor or dilatation of f Significance editThurston constructed a compactification of the Teichmuller 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 Poincare half plane A generic diffeomorphism of a surface of genus at least two is isotopic to a pseudo Anosov diffeomorphism Generalization editUsing 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 groups on the algebraic side This leads to an analogue of Thurston classification for the case of automorphisms of free groups developed by Bestvina and Handel References editA Casson S Bleiler Automorphisms of Surfaces after Nielsen and Thurston London Mathematical Society Student Texts 9 1988 A Fathi F Laudenbach and V Poenaru 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 Retrieved from https en wikipedia org w index php title Pseudo Anosov map amp oldid 1020249123, wikipedia, wiki, book, books, library,

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