Adapted metrics for singular hyperbolic flows

TL;DR

This paper proves the existence of specialized metrics adapted to singular hyperbolic sets and extends the concept of adapted metrics to broader classes of hyperbolic flows with singularities.

Contribution

It introduces singular and 2-sectional adapted metrics for singular hyperbolic sets, expanding the classical hyperbolic theory to include singularities.

Findings

Existence of singular adapted metrics for any singular hyperbolic set.

Existence of 2-sectional adapted metrics for certain classes of hyperbolic sets.

Extension of adapted metric concepts to flows with singularities.

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.