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Expansive homeomorphisms and hyperbolic diffeomorphisms on 3-manifolds

Published online by Cambridge University Press:  19 September 2008

José L. Vieitez
José L. Vieitez
Affiliation:
Inst. de Matemática, Fac. de Ingenieria, Universidad de la República, Montevideo, Uruguay
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Abstract

This paper is a contribution to the classification problem of expansive homeomorphisms. Let M be a compact connected oriented three dimensional topological manifold without boundary and f: M → M an expansive homeomorphism.We show that if the topologically hyperbolic period points of f are dense in M then M = , and f is conjugate to an Anosov diffeomorphism. This follows from our basic result: for such a homeomorphism, all stable and unstable sets are (tamely embedded) topological manifolds.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 3 , June 1996 , pp. 591 - 622
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[B]Bing, R. H.. The Geometric Topology of 3-Manifolds. Amer. Math. Soc. Coll. Pub., vol. 40 (1983).Google Scholar
[D-H]Duvall, P. F. and Husch, L. S.. Analysis on topological manifolds. Fund. Math. LXXVD (1972), 75–90.CrossRefGoogle Scholar
[Fa]Falconer, K. J.. The Geometry of Fractal Sets. Cambridge University Press, Cambridge, 1985.CrossRefGoogle Scholar
[Fr]Franks, J.. Anosov diffeomorphisms. Global Analysis Proc. Symp. Pure Math 14. Amer. Math. Soc., pp. 61–93 (1970).CrossRefGoogle Scholar
[F-R]Franks, J. and Robinson, R. C.. A quasi-Anosov diffeomorphism that is not Anosov. Trans. Amer. Math. Soc. 233 (1976), 267–278.CrossRefGoogle Scholar
[H1]Hiraide, K.. Expansive homeomorphisms with the pseudo-orbit tracing property of n-tori. J. Math. Soc. Japan 41 (3) (1989), 357–389.Google Scholar
[H2]Hiraide, K.. Expansive homeomorphisms of compact surfaces are pseudo-Anosov. Osaka J. Math. 27 (1990), 117–162.Google Scholar
[H-H]Hector, G. and Hirsch, U.. Introduction to the Geometry of Foliations, Part B, Foliations of Codimension One, Aspects of Mathematics, Vol E3. Vieweg, 1983.CrossRefGoogle Scholar
[H-W]Hurewicz, W. and Wallman, H.. Dimension Theory (Princeton Math. Series 4). Princeton University Press, 1948.Google Scholar
[K]Kuratowski, K.. Topology. Academic, New York-London, 1966.Google Scholar
[L1]Lewowicz, J.. Persistence in expansive systems. Ergod. Th. & Dynam. Sys. 3 (1983), 567–578.CrossRefGoogle Scholar
[L2]Lewowicz, J.. Expansive homeomorphisms of surfaces. Bol. Soc. Bras. Mat. 20 (1) (1989), 113–133.CrossRefGoogle Scholar
[M]Moise, E. E.. Geometric Topology in Dimensions 2 and 3. Springer, New York, 1977.CrossRefGoogle Scholar
[R]Roussarie, R.. Sur les Feuilletages des Variéetés de dimension trois. Ann. Inst. Fourier, Grenoble 21 (3) (1971), 13–82.CrossRefGoogle Scholar
[S]Spanier, E. H.. Algebraic Topology. McGraw-Hill, New York, 1965.Google Scholar
[V1]Vieitez, J. L.. Three Dimensional Expansive Homeomorphisms, Dynamical Systems. Santiago de Chile 1990, Pitman Series, Harlow, 1993.Google Scholar
[V2]Vieitez, J. L.. Funciones de Liapunov Hiperbó1icas. Publicaciones Matem. del Uruguay. 27–35Google Scholar
[W]Wilder, R. L.. Topology of Manifolds. AMS, Providence, RI, 1976.Google Scholar