In this talk, we will present the renormalization operator for multicritical circle maps and we will give certain conditions to get rigidity results, for almost all irrational rotation number. This is a joint work with Pablo Guarino (UFF-Brazil).
We extend to open surfaces with simple singularities the theorem proposed by Xavier Jarque and Zbigniew Nitecki for Hamiltonian flows in the plane that are structurally stable among Hamiltonian flows. We describe the Hamiltonian dynamics on open surfaces by presenting characterization theorems for Hamiltonian stability and some natural consequences.
We show that any smooth area-preserving diffeomorphism of a two-dimensional surface which has an elliptic fixed point can be perturbed to one exhibiting a chaotic island whose metric entropy is positive, the perturbation is small in the C-infinity topology. This proves a conjecture by Herman stating that the identity map of the disk can be perturbed to a conservative diffeomorphism with positive metric entropy. This implies also that the Chirikov standard map for large and small parameter values can be arbitrarily well approximated by a conservative diffeomorphism with a positive metric entropy (a weak version of Sinai’s positive metric entropy conjecture). We argue that these results question the relevance of the notions of metric entropy and Lyapunov exponents to the grand problem of understanding the Hamiltonian chaos. This is a joint work with Pierre Berger.
In this talk I will present local classification results for all typical singularities of Hamiltonian systems with one-sided constraints, e.g. a boundary or an obstacle lifted from the configuration space of the system to its phase space. In particular I will show how to obtain exact normal forms with functional moduli, and how to give a canonical -geometric- description of these moduli in terms of discriminant functions in the space of all solutions of the corresponding Hamiltonian system.
In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages. The most useful tool in this area is the celebrated Mañé Lemma, in its various forms. In this talk, we discuss a non-commutative Mañé Lemma, suited to the problem of maximization of Lyapunov exponents of linear cocycles or, more generally, vector bundle automorphisms. More precisely, we provide conditions that ensure the existence of an extremal norm, that is, a Finsler norm with respect to which no vector can be expanded in a single iterate by a factor bigger than the maximal asymptotic expansion rate. This is a joint work with Jairo Bochi (Pontificia Universidad Católica de Chile).
We study Markovian random iterations of homeomorphisms on the circle. Recently, Malicet stablished an invariance principle for random iterations of homeomorphisms on the circle over non-invertible dynamical systems and use it to obtain local synchronization in the i.i.d. case. We will present a version of the invariance principle for random iterations over invertible dynamical systems and show how to use it to obtain local synchronization in the Markovian case.
Sua pesquisa na área de redes complexas de sistemas interagentes foi premiada pelo programa The Newton Advanced Fellowship, da The Royal Society. Com sede em Londres, a instituição acadêmica é uma das mais antigas e renomadas do mundo, fundada em 1660, tendo entre seus presidentes Isaac Newton.
