Infinitesimal Lyapunov functions for singular flows

Abstract

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

Singular hyperbolicity and sectional Lyapunov exponents of various orders

“…It is also given a characterization of singular and sectional hyperbolicity for a flow over a compact invariant set, improving a result in [1].…”

“…Our first main result concerns in a characterization of singular hyperbolicity of any order via infinitesimal Lyapunov functions, following [1], [2], [11], [21], [23], [25].…”

Adapted Metrics for Codimension One Singular Hyperbolic Flows

“…values in E. Let J be a C 1 family of indefinite quadratic forms such that A t (x) is strictly J-separated. Then For the proof and more details about the Proposition 3.4, see [6]. Above we writeJ(v) =<J x v, v >, whereJ x is given in Proposition 2.1, that is,J(v) is the time derivative of J under the action of the flow.…”

“…An example of application of the adapted metric from [15] is contained in [6], where the first author jointly with V. Araújo, following the spirit of Lewowicz's result, construct quadratic forms which characterize partially hyperbolic and singular hyperbolic structures on a trapping region for flows.…”

“…[6, Proposition 1.3] A J-non-negative vector field X on U is strictly J-separated if, and only if, there exists a compatible family J 0 of forms and there exists a function δ : U → R such that the operatorJ 0,x :…”