Sectional Lyapunov exponents

2018

Proc. Amer. Math. Soc.

It is given notions of singular hyperbolicity and sectional Lyapunov exponents of orders beyond the classical ones, namely, other dimensions besides the dimension 2 and the full dimension of the central subbundle of the singular hyperbolic set. It is obtained a characterization of dominated splittings, partial and singular hyperbolicity in this broad sense, by using Lyapunov exponents and the notion of infinitesimal Lyapunov functions. Furthermore, it is given alternative requirements to obtain singular hyperbolicity. As an application we obtain some results related to singular hyperbolic sets for flows.2000 Mathematics Subject Classification. Primary: 37D30; Secondary: 37D25.

Section: P-sectional Lyapunov Exponentsmentioning

confidence: 97%

Singular hyperbolicity and sectional Lyapunov exponents of various orders

2018

Proc. Amer. Math. Soc.

“…The subadditive ergodic theorem (see [15]) shows that the function f (x) = lim inf t→+∞ [5,Proposition 3.4].) Let {t → f t : S → R} t∈R be a continuous family of continuous functions which is sub-additive and suppose that f (x) < 0 in a set of total probability.…”

Section: Proof Of Theorem 27mentioning

confidence: 99%

“…By a sectional hyperbolic set we mean a compact and invariant set of a vector field in a manifold (possibly with boundary) such that the tangent bundle on it has a dominated splitting, where the area along any two-planes inside one of subbundles is uniformly expanded, any vector along the other subbundle is uniformly contracted and all of its singularities are hyperbolic or for −X. There is a recent great interest to understand these flows and its dynamical properties, see [5,6,3,4,8,19], for instance.…”

Section: Introductionmentioning

confidence: 99%

“…In [5], the first author gives a characterization of sectional-Anosov flows based on Lyapunov exponents and sectional Lyapunov exponents. The latter are defined using the second exterior power of the derivative cocycle.…”

Section: Introductionmentioning

confidence: 99%

“…Namely, we define nonuniformly sectional hyperbolic sets. Using the ideas and some results from [5] and the techniques from [11], we prove that if an open subset U of the space X 1 (M) of C 1 -vector fields contains a residual subset S whose elements have a nonuniformly sectional hyperbolic critical set, then U has a residual subset whose elements are vector fields with sectional hyperbolic nonwandering set. Actually, we also have another reformulation of our result in terms of residual sets.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On critical orbits and sectional hyperbolicity of the nonwandering set for flows

Arbieto

Journal of Differential Equations

2011

Self Cite

MSC: 37C10 37D2Keywords: Nonuniformly sectional hyperbolic set Sectional hyperbolic sets Mañé ergodic general density theorem In this note, we introduce the notion of nonuniformly sectional hyperbolic set and use it to prove that any C 1 -open set which contains a residual subset of vector fields with nonuniformly sectional hyperbolic critical set also contains a residual subset of vector fields with sectional hyperbolic nonwandering set. This not only extends Theorem A of Castro [11], but using suspensions we recover it.

Adapted Metrics for Codimension One Singular Hyperbolic Flows

New Trends in One-Dimensional Dynamics

Coelho

2019

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.