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ByEdmilson Santos

On positive metric entropy problem

Dmitry Turaev

Abstract:

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.

ByEdmilson Santos

Local Normal Forms of Hamiltonian Systems with One-Sided Constraints

Konstantinos Kourliouros

Abstract:

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.

ByEdmilson Santos

Extremal norms for fiber-bunching cocycles

Eduardo Garibaldi

Abstract:

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).

ByEdmilson Santos

Synchronization of Markovian random iterations of homeomorphisms on the circle

Edgar Matias da Silva

Abstract:

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.

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