The Lyapunov exponents of generic zero divergence three-dimensional vector fields

Abstract

We prove that for a C1-generic (dense Gδ) subset of all the conservative vector fields on three-dimensional compact manifolds without singularities, we have for Lebesgue almost every (a.e.) point p∈M that either the Lyapunov exponents at p are zero or X is an Anosov vector field. Then we prove that for a C1-dense subset of all the conservative vector fields on three-dimensional compact manifolds, we have for Lebesgue a.e. p∈M that either the Lyapunov exponents at p are zero or p belongs to a compact invariant set with dominated splitting for the linear Poincaré flow.

Dominated splitting and zero volume for incompressible three flows

“…Finally using standard arguments from Bochi-Viana [14], [9] and [10] these results imply the C 1 generic dichotomies mentioned above for incompressible flows without any extra condition on the singularities.…”

“…Recently (see Theorem 1.1 below) one of the coauthors was able to use, adapt and fully extend the ideas of the original proof by Bochi to the setting of generic conservative flows on three-dimensional compact boundaryless manifolds without singularities, in [9]. The presence of singularities imposes some differences between discrete and continuous systems.…”

“…The presence of singularities imposes some differences between discrete and continuous systems. The ideas from the Bochi-Mañé proof were partially extended to a dense subset of C 1 incompressible flows (see Theorem 1.2 below) admitting singularities but without a full dichotomy between zero exponents and global hyperbolicity in the same work [9].…”

A generic incompressible flow is topological mixing

“…We consider the tiled cube (of a fixed chart based at x) C := ϕ −1 (C ϕ ) ⊂ Σ where Σ is an (n − 1)-dimensional submanifold transversal to the flow direction. Given N 1 we say that {x i } n i=0 preserves the tiling in the flowbox 3 for j = 1, . .…”

“…Roughly speaking S s moves a given point (x, r) to (x, r + s) at velocity one until hits the graph of h, after that the point returns to the transversal section Σ (base). It is well known that any aperiodic 4 flow is equivalent to some special flow (see [3] and the references therein). In [3, Section 3.6.1] we use this fact to build a Kakutani castle with very high towers in order to avoid overlapping of the local perturbations and, moreover, to have enough time to perform lots of small perturbations.…”