Adapted metrics for singular hyperbolic flows

Abstract

Singular and sectional hyperbolic sets are the objects of the extension of the classical Smale Hyperbolic Theory to flows having invariant sets with singularities accumulated by regular orbits within the set. It is by now well-known that (partially) hyperbolic sets admit adapted metrics. We show the existence of singular adapted metrics for any singular hyperbolic set with respect to a $C^{1}$ vector field on finite dimensional compact manifolds. Moreover, we obtain 2-sectional adapted metrics for certain open classes of 2-sectional hyperbolic sets and also for any hyperbolic set.