Araujo, V., Salgado, L. S. - Dominated Splittings for exterior powers and singular hyperbolicity
Abstract. We relate dominated splitting for a linear multiplicative cocyle with dominated splitting for the exterior powers of this cocycle. For a C1 vector field X on a 3-manifold, we can 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. In this setting, we also improve a result from [V. Araujo and L. Salgado. Infinitesimal lyapunov functions for singular flows. Mathematische Zeitschrift (online), pages 1–35, 2013]. As a consequence, we show the existence of adapted metrics for singular-hyperbolic sets for three-dimensional C1 vector fields.