Centre of Mathematics of the University of Porto

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Lyapunov exponents for linear homogeneous differential equations

Publication . Bessa, Mário

We 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.

2023Conference object

Restricted access

Simple Lyapunov spectrum for linear homogeneous differential equations with Lp parameters

Publication . Amaro, Dinis; Bessa, Mário; Vilarinho, Helder

In 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.

2024Journal article

Open access

Billiards in generic convex bodies have positive topological entropy

Publication . Bessa, Mário; Del Magno, Gianluigi; Dias, João Lopes; Gaivão, José Pedro; Torres, Maria Joana

We show that there exists a C2-open dense set of convex bodies with smooth boundary whose billiard map exhibits a non-trivial hyperbolic basic set. As a consequence billiards in generic convex bodies have positive topological entropy and exponential growth of the number of periodic orbits.

2024Journal article

Open access

Genericity of trivial Lyapunov spectrum for L-cocycles derived from second order linear homogeneous differential equations

Publication . Amaro, Dinis; Bessa, Mário; Vilarinho, Helder

We 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.

2024Journal article

Open access