Adapted Metrics for Codimension One Singular Hyperbolic Flows

Abstract

For a partially hyperbolic splitting T Γ M = E ⊕ F of Γ, a C 1 vector field X on a m-manifold, we obtain singular-hyperbolicity using only the tangent map DX of X and its derivative DX t whether E is one-dimensional subspace. We show the existence of adapted metrics for singular hyperbolic set Γ for C 1 vector fields if Γ has a partially hyperbolic splitting T Γ M = E⊕F where F is volume expanding, E is uniformly contracted and a one-dimensional subspace.

Star flows: a characterization via Lyapunov functions

“…More results relating geometric and algebraic features of singular hyperbolicity can be viewed in [5], [6], [36], for the classical sectional and singular hyperbolicity definitions. Also see [38] for singular hyperbolicity in a broad sense involving sectional expansion of intermediate dimensions between two and the full dimension of the central subbundle.…”

Adapted metrics for singular hyperbolic flows

“…We stress that the contructions of adapted metrics in [4,14], via quadratic forms, is deeply based on the dimension of the singular hyperbolic subbundles. Thus, it is not clear how to use quadratic forms to obtain adapted metrics when the codimension between the p-sectional hyperbolic splitting is not equal to one.…”

“…In [14,Theorem B], the second and last authors showed the existence of adapted metrics for any singular hyperbolic set Γ of a C 1 vector fields in the particular setting where Γ has a partially hyperbolic splitting T Γ M = E ⊕ F with F volume expanding and E an one-dimensional uniformly contracting bundle, extending the result from [4] for any codimension one singular hyperbolic set. This is also done under the point of view of J-algebras of Potapov [20], confirming the very interesting feature of the quadratic forms technique from which we can get adapted metrics.…”

“…Namely, we prove the existence of adapted metrics for any singular hyperbolic (of any codimension) set Γ for C 1 vector fields, but in this work this is made in a certain different way from [4,14]. Now, we make this without using quadratic forms, we only use multilinear algebra and the dynamics.…”