# 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, 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.  Google Scholar [2] A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Commun. Math. Phys., 126 (1989), 249-262. 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), 217-226.  Google Scholar [4] Jorge Groisman, "Expansive Homeomorphisms of the Plane," Ph.D. thesis, Universidad de la República, Uruguay, 2007. Google Scholar [5] J. Groisman, Expansive homeomorphisms of the plane, Discrete and Continuous Dynam. Systems, 29 (2011), 213-239.  Google Scholar [6] K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math., 27 (1990), 117-162.  Google Scholar [7] K. Kuratowski, "Topology," Academic Press, New York-London, 1966. Google Scholar [8] J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Bras. Mat. (N.S.), 20 (1989), 113-133.  Google Scholar [9] W. White, An Anosov translation, in "Dynamical Systems" (Proc. Sympos., Univ. of Bahia, Salvador, 1971), Academic Press, New York, (1973), 667-670.  Google Scholar

show all references

##### References:
 [1] L. Brouwer, Beweis des ebenen Translationssatzes, Math. Ann., 72 (1912), 37-54. doi: 10.1007/BF01456888.  Google Scholar [2] A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Commun. Math. Phys., 126 (1989), 249-262. 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), 217-226.  Google Scholar [4] Jorge Groisman, "Expansive Homeomorphisms of the Plane," Ph.D. thesis, Universidad de la República, Uruguay, 2007. Google Scholar [5] J. Groisman, Expansive homeomorphisms of the plane, Discrete and Continuous Dynam. Systems, 29 (2011), 213-239.  Google Scholar [6] K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math., 27 (1990), 117-162.  Google Scholar [7] K. Kuratowski, "Topology," Academic Press, New York-London, 1966. Google Scholar [8] J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Bras. Mat. (N.S.), 20 (1989), 113-133.  Google Scholar [9] W. White, An Anosov translation, in "Dynamical Systems" (Proc. Sympos., Univ. of Bahia, Salvador, 1971), Academic Press, New York, (1973), 667-670.  Google Scholar
 [1] Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399 [2] Jorge Groisman. Expansive homeomorphisms of the plane. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 213-239. doi: 10.3934/dcds.2011.29.213 [3] Alfonso Artigue. Anomalous cw-expansive surface homeomorphisms. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3511-3518. doi: 10.3934/dcds.2016.36.3511 [4] Alfonso Artigue. Lipschitz perturbations of expansive systems. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1829-1841. doi: 10.3934/dcds.2015.35.1829 [5] Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231 [6] El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449 [7] Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355 [8] Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447 [9] John Erik Fornæss. Sustainable dynamical systems. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1361-1386. doi: 10.3934/dcds.2003.9.1361 [10] Vieri Benci, C. Bonanno, Stefano Galatolo, G. Menconi, M. Virgilio. Dynamical systems and computable information. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 935-960. doi: 10.3934/dcdsb.2004.4.935 [11] Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91 [12] Josiney A. Souza, Tiago A. Pacifico, Hélio V. M. Tozatti. A note on parallelizable dynamical systems. Electronic Research Announcements, 2017, 24: 64-67. doi: 10.3934/era.2017.24.007 [13] Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1 [14] Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432 [15] Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587 [16] Alexander Sakhnovich. Dynamical canonical systems and their explicit solutions. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1679-1689. doi: 10.3934/dcds.2017069 [17] Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215 [18] Jérôme Rousseau, Paulo Varandas, Yun Zhao. Entropy formulas for dynamical systems with mistakes. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4391-4407. doi: 10.3934/dcds.2012.32.4391 [19] Giuseppe Gaeta. On the geometry of twisted prolongations, and dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1209-1227. doi: 10.3934/dcdss.2020070 [20] Yujun Zhu. Preimage entropy for random dynamical systems. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829

2019 Impact Factor: 1.338