Infinitesimal Lyapunov functions for singular flows
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Mathematische Zeitschrift
Infinitesimal Lyapunov functions for singular flows Vitor Araujo · Luciana Salgado
Received: 15 June 2012 / Accepted: 27 February 2013 © Springer-Verlag Berlin Heidelberg 2013
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 D X 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. Keywords Dominated splitting · Partial hyperbolicity · Sectional hyperbolicity · Infinitesimal Lyapunov function Mathematics Subject Classification (2000)
Primary 37D30; Secondary 37D25
1 Introduction The hyperbolic theory of dynamical systems is now almost a classical subject in mathematics and one of the main paradigms in dynamics. Developed in the 1960s and 1970s after the work of Smale, Sinai, Ruelle, Bowen [11,12,42,43], among many others, this theory deals
V.A. was partially supported by CAPES, CNPq, PRONEX-Dyn.Syst. and FAPERJ (Brazil). L.S. was supported by a CNPq doctoral scholarship and is presently supported by a INCTMat-CAPES post-doctoral scholarship. V. Araujo (B) Instituto de Matemática, Universidade Federal da Bahia, Av. Adhemar de Barros, S/N, Ondina, 40170-110 Salvador, BA, Brazil e-mail: [email protected] L. Salgado Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, Jardim Botânico, Rio de Janeiro 22460-320, Brazil e-mail: [email protected]
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with compact invariant sets for diffeomorphisms and flows of closed finite-dimensional manifolds having a hyperbolic splitting of the tangent space. That is, T M = E s ⊕ E X ⊕ E u is a continuous splitting of the tangent bundle over , where E X is the flow direction, the subbundles are invariant under the derivative D X t of the flow X t D X t · E x∗ = E ∗X t (x) , x ∈ , t ∈ R, ∗ = s, X, u; E s is uniformly contracted by D X t and E u is likewise expanded: there are K , λ > 0 so that D X t | E xs ≤K e−λt , (D X t | E xu )−1 ≤K e−λt , x ∈ , t ∈ R. Very strong properties can be deduced from the existence of such hyperbolic structure; see for instance [11,12,18,37,41]. More recently, extensions of this theory based on weaker notions of hyperbolicity, like the notions of dominated splitting, partial hyperbolicity, volume hyperbolicity and singular hyperbolicity (for three-dimensional flows) have been developed to encompass larger classes of systems beyond the uniformly hyperbolic ones; see [7] and specifically [5,47] for singular hyperbolicity and Lorenz-like attractors. One of the technical difficulties in this theory is to actually prove the existence of a hyperbolic structure, even in its weaker forms. We mention that Malkus showed that the Lorenz equatio
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