Dominated splitting for exterior powers and singular hyperbolicity

Abstract

We show that an invariant splitting for the tangent map to a smooth flow over a compact invariant subset is dominated if, and only if, the exterior power of the tangent map admits an invariant dominated splitting. For a C1 vector field Xon a 3-manifold, we obtain singular-hyperbolicity using only the tangent map DX of X and a family of indefinite and non-degenerate quadratic forms without using the associated flow Xt and its derivative DXt. As a consequence, we show the existence of adapted metrics for singular-hyperbolic sets for three-dimensional C1 vector fields.