We present an extension of the notion of infinitesimal Lyapunov function to singular flows, and from this technique we deduce a characterization of partial/sectional hyperbolic sets. In absence of singularities, we can also characterize uniform hyperbolicity. These conditions can be expressed using the space derivative DX of the vector field X together with a field of infinitesimal Lyapunov functions only, and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region.Comment: 37 pages, 1 figure; corrected the statement of Lemma 2.2 and item (2) of Theorem 2.7; removed item (5) of Theorem 2.7 and its wrong proof since the statement of this item was false; corrected items (1) and (2) of Theorem 2.23 and their proofs. Included Example 6 on smooth reduction of families of quadratic forms. The published version in Math Z journal needs an errat

Abstract. We relate dominated splitting for a linear multiplicative cocyle with dominated splitting for the exterior powers of this cocycle. For a C 1 vector field X on a 3-manifold, we can obtain singular-hyperbolicity using only the tangent map DX of X and a family of indefinite and non-degenerate quadratic forms without using the associated flow X t and its derivative DX t . In this setting, we also improve a result from [6]. As a consequence, we show the existence of adapted metrics for singular-hyperbolic sets for three-dimensional C 1 vector fields.

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.

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)