Dominated splitting and zero volume for incompressible three-flows

Details Dominated splitting and zero volume for incompressible three-flows Dominated splitting and zero volume for incompressible three-flows

2008-01-14

arxiv.org/abs/0801.2148v2

Nonlinearity 21 (2008), 1637-1653

Mathematics

Dynamical Systems

23 pages, no figures

Scientific paper

10.1088/0951-7715/21/7/014

We prove that there exists an open and dense subset of the incompressible 3-flows of class C^2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincar\'e flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi-Ma\~n\'e and of Newhouse for flows with singularities. That is we obtain for a residual subset of the C^1 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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