Adapted Metrics for Codimension one Singular Hyperbolic Flows

TL;DR

This paper develops adapted metrics for singular hyperbolic sets with a one-dimensional contracting bundle, extending previous results to include cases with singularities and volume-expanding subbundles.

Contribution

It introduces new adapted metrics for singular hyperbolic sets with one-dimensional bundles, broadening the scope of existing hyperbolic theory to include singularities.

Findings

Existence of adapted metrics for singular hyperbolic sets with one-dimensional bundles.

Extension of previous results to include singularities and volume-expanding subbundles.

Generalization of hyperbolic theory to broader classes of dynamical systems.

Abstract

For a partially hyperbolic splitting a $C^1$ vector field $X$ on a $m$-manifold $M$, we obtain singular hyperbolicity whether $E$ is one-dimensional subspace, based on the idea of cross products. We show the existence of adapted metrics for singular hyperbolic set if it has a partially hyperbolic splitting $T_{\Gamma}M = E \oplus F$, where $F$ is a volume expanding subbundle, $E$ is an uniformly contracted and one-dimensional subbundle. Theses results extend previous ones from the first author and V. Ara\'ujo.