# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. doi: 10.1007/BF01456888. Google Scholar [2] A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension,, Commun. Math. Phys., 126 (1989), 249. doi: 10.1007/BF02125125. 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] J. Groisman, Expansive homeomorphisms of the plane,, Discrete and Continuous Dynam. Systems, 29 (2011), 213. Google Scholar [6] K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov,, Osaka J. Math., 27 (1990), 117. Google Scholar [7] K. Kuratowski, "Topology,", Academic Press, (1966). Google Scholar [8] J. Lewowicz, Expansive homeomorphisms of surfaces,, Bol. Soc. Bras. Mat. (N.S.), 20 (1989), 113. Google Scholar [9] W. White, An Anosov translation,, in, (1973), 667. Google Scholar

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##### References:
 [1] L. Brouwer, Beweis des ebenen Translationssatzes,, Math. Ann., 72 (1912), 37. doi: 10.1007/BF01456888. Google Scholar [2] A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension,, Commun. Math. Phys., 126 (1989), 249. doi: 10.1007/BF02125125. 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] J. Groisman, Expansive homeomorphisms of the plane,, Discrete and Continuous Dynam. Systems, 29 (2011), 213. Google Scholar [6] K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov,, Osaka J. Math., 27 (1990), 117. Google Scholar [7] K. Kuratowski, "Topology,", Academic Press, (1966). Google Scholar [8] J. Lewowicz, Expansive homeomorphisms of surfaces,, Bol. Soc. Bras. Mat. (N.S.), 20 (1989), 113. Google Scholar [9] W. White, An Anosov translation,, in, (1973), 667. Google Scholar
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