January  2011, 29(1): 213-239. doi: 10.3934/dcds.2011.29.213

Expansive homeomorphisms of the plane

1. 

Instituto de Matemática y Estadística "Prof. Ing. Rafael Laguardia”, Facultad de Ingeniería, Julio Herrera y Reissig 565 11300, MONTEVIDEO, Uruguay

Received  January 2009 Revised  April 2010 Published  September 2010

This article tackles the problem of the classification of expansive homeomorphisms of the plane. Necessary and sufficient conditions for a homeomorphism to be conjugate to a linear hyperbolic automorphism will be presented. The techniques involve topological and metric aspects of the plane. The use of a Lyapunov metric function which defines the same topology as the one induced by the usual metric but that, in general, is not equivalent to it is an example of such techniques. The discovery of a hypothesis about the behavior of Lyapunov functions at infinity allows us to generalize some results that are valid in the compact context. Additional local properties allow us to obtain another classification theorem.
Citation: Jorge Groisman. Expansive homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 213-239. doi: 10.3934/dcds.2011.29.213
References:
[1]

L. Brouwer, Beweis des ebenen Translationssatzes,, Math. Ann., 72 (1912), 37.  doi: doi:10.1007/BF01456888.  Google Scholar

[2]

A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces (Seminaire Orsay),, Asterisque, (1979), 66.   Google Scholar

[3]

J. Franks, A new proof of the Brouwer plane translation theorem,, Ergod. Th. and Dynamic. Sys., 12 (1991), 217.   Google Scholar

[4]

Jorge Groisman, "Expansive Homeomorphisms of the Plane,", Ph.D thesis, (2007).   Google Scholar

[5]

K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov,, Osaka J. Math., 27 (1990), 117.   Google Scholar

[6]

K. Kuratowski, "Topology,", Academic Press, (1966).   Google Scholar

[7]

J. Lewowicz, Expansive homeomorphisms of surfaces,, Bol. Soc. Bras. Mat., 20 (1989), 113.   Google Scholar

[8]

W. Thurston, On the geometry dynamics of diffeomorphisms of surfaces,, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417.   Google Scholar

show all references

References:
[1]

L. Brouwer, Beweis des ebenen Translationssatzes,, Math. Ann., 72 (1912), 37.  doi: doi:10.1007/BF01456888.  Google Scholar

[2]

A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces (Seminaire Orsay),, Asterisque, (1979), 66.   Google Scholar

[3]

J. Franks, A new proof of the Brouwer plane translation theorem,, Ergod. Th. and Dynamic. Sys., 12 (1991), 217.   Google Scholar

[4]

Jorge Groisman, "Expansive Homeomorphisms of the Plane,", Ph.D thesis, (2007).   Google Scholar

[5]

K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov,, Osaka J. Math., 27 (1990), 117.   Google Scholar

[6]

K. Kuratowski, "Topology,", Academic Press, (1966).   Google Scholar

[7]

J. Lewowicz, Expansive homeomorphisms of surfaces,, Bol. Soc. Bras. Mat., 20 (1989), 113.   Google Scholar

[8]

W. Thurston, On the geometry dynamics of diffeomorphisms of surfaces,, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417.   Google Scholar

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