Dominated splitting and zero volume for incompressible three-flows

TL;DR

This paper establishes a dichotomy for generic incompressible 3-flows, showing they are either Anosov or have zero Lyapunov exponents, extending classical results to flows with singularities.

Contribution

It proves that flows with dominated splitting and positive volume invariant sets are Anosov, extending Bochi-Ma e and Newhouse dichotomies to flows with singularities.

Findings

Generic flows are either Anosov or have zero Lyapunov exponents.

Elliptic periodic points are dense unless the flow is Anosov.

Flows with dominated splitting and positive volume sets are necessarily Anosov.

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.