Dominated splittings for flows with singularities

Abstract

We obtain sufficient conditions for an invariant splitting over a compact invariant subset of a C 1 flow Xt to be dominated. In particular, we reduce the requirements to obtain sectional hyperbolicity and hyperbolicity.Very strong properties can be deduced from the existence of a such structure; see for instance [13,25].Weaker notions of hyperbolicity, like the notions of dominated splitting, partial hyperbolicity, volume hyperbolicity and singular or sectional hyperbolicity (for singular flows) have been proposed to try to enlarge the scope of this theory to classes of systems beyond the uniformly hyperbolic ones; see [5] and [1] for singular or sectional hyperbolicity. However, the existence of dominated splittings is the weaker one.Many researchers have studied the relations of the existence of dominated splittings with other dynamical phenomena, mostly in the discrete time case, such as robust transitivity, homoclinic tangencies and heteroclinic cycles, and also the possible extension of this notion to endomorphisms; see for instance [20,33,21,19,5,6,15].However, the notion of dominated splittings deserves attention. Several authors used this notion for the Linear Poincaré flow, see [11,16,17], and this is useful in the absence of singularities, as in [12]. We remark that this flow is defined only in the set of regular points. However, in the singular case, it is a non-trivial question to obtain a dominated splitting for the derivative of the flow, and thus allowing singularities. Indeed, it is difficult to obtain it, from a dominated splitting for the Linear Poincaré Flow. See [14], for an attempt to solve this, in the context of robustly transitive sets, using the extended Linear Poincaré Flow.for every x ∈ Λ and such that there are positive constants K, λ satisfying DX t | Ex · DX −t | F X t (x) < Ke −λt , for all x ∈ Λ, and all t > 0.( 1.2)

Singular hyperbolicity and sectional Lyapunov exponents of various orders

“…However, we might ask how this assumption worked out in [2] and [3]. In fact, within the accounts of the results contained therein it is obtained domination due 2-sectional expansion together with the uniform contraction.…”

“…In [3], this author together with V. Araujo and A. Arbieto, proved that the requirements in the definition of sectional hyperbolicity can be weakened, demanding the domination property only over the singularities, because in this setting the splitting is in fact dominated. More precisely, we proved the next result.…”

“…The case of positive Lyapunov exponents over F is analogous, by taking f t (x) = log DX −t | F x . (Also see proof of [3,Theorem B]).…”

“…Let Λ be a compact invariant set for a flow X of a C 1 vector field X on M. Lemma 3.2. [3] Given a continuous splitting T Λ M = E ⊕ F such that E is uniformly contracted, then X (x) ∈ F x for all x ∈ Λ.…”

A characterization of singular-hyperbolicity

“…DX T (X t (x))| F X t (x) | dt = log | det DX T | F | dμfor μ-a.e. x, and the chain rule as in(3det DX t (x)| F x | dt det DX T (X t (x))| F X t (x) | dt, so there exists x ∈ * ∩ R satisfying lim DX t (x)| F x | dt ≤ 0. (3.13)…”