Dominated splitting for exterior powers and singular hyperbolicity

TL;DR

This paper establishes a connection between dominated splitting for linear cocycles and their exterior powers, providing new methods to analyze singular hyperbolicity in three-dimensional vector fields without relying on flow derivatives.

Contribution

It introduces a novel approach to obtain singular hyperbolicity using only the tangent map and quadratic forms, bypassing the need for flow derivatives.

Findings

Established a relation between dominated splitting and exterior powers.

Provided a method to obtain singular hyperbolicity without flow derivatives.

Proved the existence of adapted metrics for singular-hyperbolic sets.

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 X_t and its derivative DX_t. As a consequence, we show the existence of adapted metrics for singular-hyperbolic sets for three-dimensional C1 vector fields.