Expansive homeomorphisms of surfaces

  • Jorge Lewowicz


Letf be an expansive homeomorphism of a compact oriented surfaceM. We show thatS 2 does not support such anf, and thatf is conjugate to an Anosov diffeomorphism ifM=T 2, and to a pseudo-Anosov map ifM has genus ≥2. These results are consequences of our description of local stable (unstable) sets: everyxM has a local stable (unstable) set that consists of the union ofr arcs that meet only atx. For eachxM r=2, except for a finite number of points, wherer≥3.


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  1. 1.
    Epstein, D.,Curves on 2-manifolds and isotopies, Acta Mathematica115 (1966), 83–107.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Fathi, A.; Laudenbach, F. and Poenaru, V.,Travaux de Thurston sur les surfaces, (Seminaire Orsay), Asterisque No.66-67 (1979).Google Scholar
  3. 3.
    Franks, J.,Anosov diffeomorphisms, Proceedings of the Symposium in Pure Mathematics14 (1970), 61–94.MathSciNetGoogle Scholar
  4. 4.
    Gerber, M. and Katok, A.,Smoth models of Thurston pseudo-Anosov maps, Annales Scientifiques Ecole Normale Superieure15 (1982), 173–204.MATHMathSciNetGoogle Scholar
  5. 5.
    Handel, M.,Global shadowing of pseudo Anosov homeomorphisms, Ergodic Theory and Dynamical Systems5 (1985), 373–377.MATHMathSciNetGoogle Scholar
  6. 6.
    Kuratowski, K., “Topology”, Academic Press, New York-London, 1966.Google Scholar
  7. 7.
    Lewowicz, J.,Lyapunov functions and topological stability, Journal of Differential Equations38(2) (1980), 192–209.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    —,Persistence in expansive systems, Ergodic Theory and Dynamical Systems3 (1983), 567–578.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Lewowicz, J. and Lima de Sá, E.,Analytic models of pseudo Anosov maps, Ergodic Theory and Dynamical Systems6 (1986), 385–392.MATHMathSciNetGoogle Scholar
  10. 10.
    O. Brien, T. and Reddy, W.,Each compact orientable surface of positive genus admits an expansive homeomorphism, Pacific Journal Mathematics35 (1970), 737–741.MathSciNetGoogle Scholar
  11. 11.
    Thurston, W.,On the geometry and dynamics of diffeomorphisms of surfaces, Preprint.Google Scholar

Copyright information

© Sociedade Brasileira de Matemática 1989

Authors and Affiliations

  • Jorge Lewowicz
    • 1
  1. 1.Instituto de Matemática y Estadística Prof. Ing. Rafael LaguardiaMontevideoUruguay

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