Dominated splitting and zero volume for incompressible three flows

Abstract

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 Poincaré flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi-Mañé (see [26,13,9]) and of Newhouse (see [30,10]) 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.

On the Entropy of Conservative Flows

“…The following result was proved by the first author for the case of incompressible flows on three-dimensional closed manifolds without singularities (see [4,Proposition 3.2]) and then generalized for the context admitting singularities in [3,Proposition 2.2]. Recall that X is said to be an aperiodic vector field if the Lebesgue measure of the set of periodic points and singularities is zero.…”

Homoclinic tangencies versus uniform hyperbolicity for conservative 3-flows

“…In what follows, up to a smooth conservative change of coordinates Ψ 0 defined on a neighborhood U of Γ (p, τ ), we can assume that we are working on the Euclidean space R 3 , that p = 0 and that 1 X(p) X(p) = ∂ ∂x 1 = v (see [11]). Let W ⊂ R 3 be the two-dimensional vector subspace orthogonal to the unitary vector v. Given r > 0 let B r (p) denote the ball of radius r, centered at p and contained in N p = X(p) ⊥ = W .…”

“…[5].) Given > 0 and a vector field X ∈ X 4 μ (M) there exists ξ 0 = ξ 0 ( , X) such that ∀τ ∈ [1,2], for any periodic point p of period greater than 2, for any sufficient small flowbox T of Γ (p, τ ) and…”

“…where S t : N p → N X t (p) is such that S t ∈ SL(2, R) for every t ∈ R. Actually, S t has the same dynamics of the linear Poincaré flow modulo the distortion factor given by a ratio involving the norm of the vector field (see (1) and (2)). …”