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The generic symplectic C^{1}-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point

Published online by Cambridge University Press:  06 November 2002

MARIE-CLAUDE ARNAUD
Affiliation:
EA 2151, Laboratoire d'Analyse non linéaire et Géométrie, UFR Sciences, Université d'Avignon, 33, rue Louis Pasteur, 84000 Avignon, France (e-mail: marie-claude.arnaud@univ-avignon.fr)

Abstract

We prove that if (M,\omega) is a connected and compact four-dimensional symplectic manifold, there exist three open sets U_1, U_2, U_3 of {\rm Diff}^1_{\omega}(M) (for the C^1 topology) such that:

  1. U_1\cup U_2\cup U_3 is dense in {\rm Diff}^1_{\omega}(M);

  2. f\in U_1 if and only if f is Anosov and transitive;

  3. f\in U_2 if and only if f is partially hyperbolic; and

  4. f\in U_3 if and only if f has a stable completely elliptic periodic point.

Type
Research Article
Copyright
2002 Cambridge University Press

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