May  2012, 32(5): 1709-1721. doi: 10.3934/dcds.2012.32.1709

Expansive and fixed point free homeomorphisms of the plane

1. 

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

Received  November 2010 Revised  September 2011 Published  January 2012

The aim of this work is to describe the set of fixed point free homeomorphisms of the plane (preserving orientation or not) under certain expansive conditions. We find necessary and sufficient conditions for a fixed point free homeomorphism of the plane to be topologically conjugate to a translation.
Citation: Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709
References:
[1]

L. Brouwer, Beweis des ebenen Translationssatzes, Math. Ann., 72 (1912), 37-54. doi: 10.1007/BF01456888.

[2]

A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Commun. Math. Phys., 126 (1989), 249-262. doi: 10.1007/BF02125125.

[3]

J. Franks, A new proof of the Brouwer plane translation theorem, Ergod. Th. and Dynamic. Sys., 12 (1991), 217-226.

[4]

Jorge Groisman, "Expansive Homeomorphisms of the Plane," Ph.D. thesis, Universidad de la República, Uruguay, 2007.

[5]

J. Groisman, Expansive homeomorphisms of the plane, Discrete and Continuous Dynam. Systems, 29 (2011), 213-239.

[6]

K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math., 27 (1990), 117-162.

[7]

K. Kuratowski, "Topology," Academic Press, New York-London, 1966.

[8]

J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Bras. Mat. (N.S.), 20 (1989), 113-133.

[9]

W. White, An Anosov translation, in "Dynamical Systems" (Proc. Sympos., Univ. of Bahia, Salvador, 1971), Academic Press, New York, (1973), 667-670.

show all references

References:
[1]

L. Brouwer, Beweis des ebenen Translationssatzes, Math. Ann., 72 (1912), 37-54. doi: 10.1007/BF01456888.

[2]

A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Commun. Math. Phys., 126 (1989), 249-262. doi: 10.1007/BF02125125.

[3]

J. Franks, A new proof of the Brouwer plane translation theorem, Ergod. Th. and Dynamic. Sys., 12 (1991), 217-226.

[4]

Jorge Groisman, "Expansive Homeomorphisms of the Plane," Ph.D. thesis, Universidad de la República, Uruguay, 2007.

[5]

J. Groisman, Expansive homeomorphisms of the plane, Discrete and Continuous Dynam. Systems, 29 (2011), 213-239.

[6]

K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math., 27 (1990), 117-162.

[7]

K. Kuratowski, "Topology," Academic Press, New York-London, 1966.

[8]

J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Bras. Mat. (N.S.), 20 (1989), 113-133.

[9]

W. White, An Anosov translation, in "Dynamical Systems" (Proc. Sympos., Univ. of Bahia, Salvador, 1971), Academic Press, New York, (1973), 667-670.

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