Dominated splitting and zero volume for incompressible three flows

Abstract

We prove that there exists an open and dense subset of the incompressible 3-flows of class C2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincaré flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi–Mañé (see Bessa 2007 Ergod. Theory Dyn. Syst. 27 1445–72, Bochi 2002 Ergod. Theory Dyn. Syst. 22 1667–96, Mañé 1996 Int. Conf. on Dynamical Systems (Montevideo, Uruguay, 1995) (Harlow: Longman) pp 110–9) and of Newhouse (see Newhouse 1977 Am. J. Math. 99 1061–87, Bessa and Duarte 2007 Dyn. Syst. Int. J. submitted Preprint 0709.0700) for flows with singularities. That is, we obtain for a residual subset of the C1 incompressible flows on 3-manifolds that: (i) either all Lyapunov exponents are zero or the flow is Anosov and (ii) either the flow is Anosov or else the elliptic periodic points are dense in the manifold.

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