Seminar

Department of Mathematics

Monday, March 24th, 2008 at 11AM

BA3008, 40 St. George Street

DYNAMICAL SYSTEMS SEMINAR -- Transitivity and Instability in Hamiltonian Dynamics.

Meysam Nassiri

IM-UFRJ, Rio de Janeiro

In Hamiltonian (symplectic dynamics), the KAM theory gives a precise description of the dynamics of many orbits (a set with large measure but nowhere dense) in neighborhood of elliptic periodic orbits or for any small perturbation of a non-degenerate integrable Hamiltonian system.

The problem of instabilities (Arnold diffusion) in Hamiltonian dynamics stands as a counterpart of KAM theory.

We introduce a new approach to this problem by studying large robustly transitive sets. As a consequence of the constructions we show that, arbitrarily $C^r$ close to certain (nearly) integrable Hamiltonian systems with more than two degrees of freedom, there exist systems with unbounded robustly transitive sets. This proves the so-called Arnold diffusion for such systems. This is a joint work with E. Pujals.

A main machinery in this approach is the transitivity of certain iterated function systems (IFS). More recently, in a joint work with A. Koropecki, we proved that any $C^r$ generic pair of conservative surface diffeomorphisms generates a transitive IFS. This result leads to the transitivity of certain partially hyperbolic sets of $C^r$ generic symplectic diffeomorphims.