INVARIANT SUBSETS OF VOLUME HYPERBOLIC SETS

A volume hyperbolic set is a compact invariant set with a dominated splitting whose external bundles uniformly contract and expand the volume respectively [1]. Examples of volume hyperbolic sets for diffeomorphisms or flows are the hyperbolic sets, the geometric Lorenz attractor [3] and the singular horseshoe [6]. We shall prove that no invariant subset of a volume hyperbolic set of a three-dimensional flow is homeomorphic to a closed surface.

A degenerate singularity generating geometric Lorenz attractors

A degenerate vector field singularity in R3 can generate a geometric Lorenz attractor in an arbitrarily small unfolding of it. This enables us to detect Lorenz-like chaos in some families of vector fields, merely by performing normal form calculations of order 3.

Lorenz knots are prime

Lorenz knots are the periodic orbits of a certain geometrically defined differential equation in ℝ3. This is called the ‘geometric Lorenz attractor’ as it is only conjecturally the real Lorenz attractor. These knots have been studied by the author and Joan Birman via a ‘knot-holder’, i.e. a certain branched two-manifold H. To show such knots are prime we suppose the contrary which implies the existence of a splitting sphere, S2. The technique of the proof is to study the intersection S2∩H. A novelty here is that S2∩H is likewise branched.