A KAM theorem for infinite-dimensional discrete systems.
A KAM-theorem for some nonlinear partial differential equations
A new proof of the Aubry-Mather's theorem.
A note on the reducibility of linear differential equations with quasiperiodic coefficients.
Almost localization and almost reducibility
Complex one-frequency cocycles
We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other...
Computer simulation of magnetic phase portraits and geometric dynamics around piecewise rectilinear circuits.
Concavity of the Lagrangian for quasi-periodic orbits.
Locally equicontinuous dynamical systems
A new class of dynamical systems is defined, the class of “locally equicontinuous systems” (LE). We show that the property LE is inherited by factors as well as subsystems, and is closed under the operations of pointed products and inverse limits. In other words, the locally equicontinuous functions in form a uniformly closed translation invariant subalgebra. We show that WAP ⊂ LE ⊂ AE, where WAP is the class of weakly almost periodic systems and AE the class of almost equicontinuous systems....
Lower and upper bounds for the splitting of separatrices of a pendulum under a fast quasiperiodic forcing.
Most automorphisms of a hyperbolic group have very simple dynamics
On the definition of strange nonchaotic attractor
The aim of this paper is twofold. On the one hand, we want to discuss some methodological issues related to the notion of strange nonchaotic attractor. On the other hand, we want to formulate a precise definition of this kind of attractor, which is "observable" in the physical sense and, in the two-dimensional setting, includes the well known models proposed by Grebogi et al. and by Keller, and a wide range of other examples proposed in the literature. Furthermore, we analytically prove that a whole...
Periodicity of β-expansions for certain Pisot units*
Given β > 1, let TβThe iteration of this transformation gives rise to the greedy β-expansion. There has been extensive research on the properties of this expansion and its dependence on the parameter β.In [17], K. Schmidt analyzed the set of periodic points of Tβ, where β is a Pisot number. In an attempt to generalize some of his results, we study, for certain Pisot units, a different expansion that we call linear expansionwhere each ei...
Reducibility of zero curvature equations.
Réductibilité presque partout des flots fibrés quasi-périodiques à valeurs dans des groupes compacts
Renormalization of cocycles and linear ODE with almost-periodic coefficients.
Rigidity of projective conjugacy for quasiperiodic flows of Koch type
For quasiperiodic flows of Koch type, we exploit an algebraic rigidity of an equivalence relation on flows, called projective conjugacy, to algebraically characterize the deviations from completeness of an absolute invariant of projective conjugacy, called the multiplier group, which describes the generalized symmetries of the flow. We then describe three ways by which two quasiperiodic flows with the same Koch field are projectively conjugate when their multiplier groups are identical. The first...
Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles
This article is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H. Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density,...
Structure of function algebras on foliated manifolds.
Structure of group invariants of a quasiperiodic flow.