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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.
- Dynamics of generic multidimensional linear differential systemsPublication . Bessa, MárioWe prove that there exists a residual subset R (with respect to the C^0 topology) of d-dimensional linear differential systems based in a μ-invariant flow and with transition matrix evolving in GL(d, R) such that if A ∈ R, then, for μ-a.e. point, the Oseledets splitting along the orbit is dominated (uniform projective hyperbolicity) or else the Lyapunov spectrum is trivial. Moreover, in the conservative setting, we obtain the dichotomy: dominated splitting versus zero Lyapunov exponents.
- Perturbations of mathieu equations with parametric excitation of large periodPublication . Bessa, MárioWe consider a linear differential system of Mathieu equations with periodic coefficients over periodic closed orbits and we prove that, arbitrarily close to this system, there is a linear differential system of Hamiltonian damped Mathieu equa- tions with periodic coefficients over periodic closed orbits such that, all but a finite number of closed periodic coefficients, have unstable solutions. The perturbations will be performed in the periodic coefficients.