João Lopes Dias

Multidimensional Continued Fractions, Dynamical Renormalization and KAM Theory

AbstractThe disadvantage of ‘traditional’ multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space

Renormalization of multidimensional Hamiltonian flows

SL(d, \mathbb{Z}) \backslash SL(d, \mathbb{R})

HAMILTONIAN ELLIPTIC DYNAMICS ON SYMPLECTIC 4-MANIFOLDS

(the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. As an example, we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension.

Realization of tangent perturbations in discrete and continuous time conservative systems

We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C2-residual set of Hamiltonians for which there is an open mod 0 dense set of regular energy surfaces each either being Anosov or having zero Lyapunov exponents almost everywhere. This is in the spirit of the Bochi-Mañé dichotomy for area-preserving diffeomorphisms on compact surfaces [2] and its continuous-time version for 3-dimensional volume-preserving flows [1].

ERGODICITY OF POLYGONAL SLAP MAPS

We construct a renormalization operator acting on the space of analytic Hamiltonians defined on , d ≥ 2, based on the multidimensional continued fractions algorithm developed by the authors. We show convergence of orbits of the operator around integrable Hamiltonians satisfying a non-degeneracy condition. This in turn yields a new proof of a KAM-type theorem on the stability of diophantine invariant tori.

Hamiltonian suspension of perturbed Poincaré sections and an application

The purpose of this paper is to study the dynamics of a square billiard with a non-standard reflection law such that the angle of reflection of the particle is a linear contraction of the angle of incidence. We present numerical and analytical arguments that the nonwandering set of this billiard decomposes into three invariant sets, a parabolic attractor, a chaotic attractor, and a set consisting of several horseshoes. This scenario implies the positivity of the topological entropy of the billiard, a property that is in sharp contrast with the integrability of the square billiard with the standard reflection law.

On the Herman–Avila–Bochi formula for Lyapunov exponents of {\rm SL}(2,\mathbb{R}) -cocycles

We consider C 2 -Hamiltonian functions on compact 4-dimensional symplectic manifolds to study the elliptic dynamics of the Hamiltonian flow, namely the so-called Newhouse dichotomy. We show that for any open set U intersecting a far from Anosov regular energy surface, there is a nearby Hamiltonian having an elliptic closed orbit through U. Moreover, this implies that, for far from Anosov regular energy surfaces of a C 2 -generic Hamiltonian, the elliptic closed orbits are generic.

Generic Hamiltonian Dynamical Systems: An Overview

We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a