Dominated splitting and zero volume for incompressible three-flows
Dynamical Systems
2009-11-13 v2 Classical Analysis and ODEs
Abstract
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.
Cite
@article{arxiv.0801.2148,
title = {Dominated splitting and zero volume for incompressible three-flows},
author = {Vitor Araujo and Mario Bessa},
journal= {arXiv preprint arXiv:0801.2148},
year = {2009}
}
Comments
23 pages, no figures