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- Genericity of trivial Lyapunov spectrum for L-cocycles derived from second order linear homogeneous differential equationsPublication . Amaro, Dinis; Bessa, Mário; Vilarinho, HelderWe consider a probability space M on which an ergodic flow is defined. We study a family of continuous-time linear cocycles, referred to as kinetic, that are associated with solutions of the second-order linear homogeneous differential equation . Our main result states that for a generic subset of kinetic continuous-time linear cocycles, where generic means a Baire second category with respect to an -like topology on the infinitesimal generator, the Lyapunov spectrum is trivial.
- Lyapunov exponents for linear homogeneous differential equationsPublication . Bessa, MárioWe consider linear continuous-time cocycles induced by second order linear homogeneous differential equations, where the coefficients evolve along the orbit of a flow defined on a closed manifold M. We are mainly interested in the Lyapunov exponents associated to most of the cocycles chosen when one allows variation of the parameters. The topology used to compare perturbations turn to be crucial to the conclusions.
- Simple Lyapunov spectrum for linear homogeneous differential equations with Lp parametersPublication . Amaro, Dinis; Bessa, Mário; Vilarinho, HelderIn the present paper we prove that densely, with respect to an Lp-like topology, the Lyapunov exponents associated to linear continuous-time cocycles induced by second order linear homogeneous differential equations are almost everywhere distinct. The coefficients evolve along the orbit for an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation and for a Schrödinger equation.
- Fine properties of Lp-cocycles which allow abundance of simple and trivial spectrumPublication . Bessa, Mário; Vilarinho, HelderIn this paper we prove that the class of accessible and saddle-conservative cocycles (a wide class which includes cocycles evolving in GL(d,R), SL(d,R) and Sp(d,R)) Lp-densely have a simple spectrum. We also prove that for an Lp-residual subset of accessible cocycles we have a one-point spectrum. Finally, we show that the linear differential system versions of previous results also hold and give some applications.
- Trivial and simple spectrum for SL(d, ℝ) cocycles with free base and fiber dynamicsPublication . Bessa, Mário; Varandas, PauloLet AC_D(M,SL(d,R)) denote the pairs (f,A) so that f ∈ A ⊂ Diff (M) is a C^1-Anosov transitive diffeomorphisms and A is an SL(d,R) cocycle dominated with respect to f. We prove that open and densely in AC_D(M,SL(d,R)), in appropriate topologies, the pair (f,A) has simple spectrum with respect to the unique maximal entropy measure μ_f. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in Aut_Leb(M) × Lp(M,SL(d,R)).