Special Year 2011-12: Symplectic Dynamics

Katok has asked the following informal question: In low dimensions are all conservative dynamical systems with zero topological entropy a limit of integrable systems? In this talk I will discuss an approach to this question that uses pseudo...

I will talk about the following theorem: If $f$ is a $C^1$ generic symplectic diffeomorphism of a compact manifold then the Oseledets splitting along almost every orbit is partially hyperbolic or trivial; in addition, if $f$ is not Anosov then all...

Recently two new resonance identities on closed characteristics on every compact star-shaped hypersurface $\Sigma$ in $R^{2n}$ are proved, when the number of geometrically distinct closed characteristics on $\Sigma$ is finite. These identities...

In this talk we discuss Hamiltonian diffeomorphisms with finitely many periodic orbits. We show that, under suitable additional hypotheses on the ambient manifold, the actions and mean indexes of periodic orbits of such a diffeomorphism must satisfy...

Consider the classical Newtonian 3-body problem, namely, bodies are mutually attracted by the Newton graviation. Call motion oscillatory if as time tends to infinity limsup of maximal distance among the bodies is infinite, while liminf is finite. In...

Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching the critical value) the corresponding diffeomorphism has a...

This is a joint work with Renato Iturriaga.

We consider a Hamiltonian $H: \mathbf R^n \times \mathbf R \to \mathbf R, \ (x,p) \mapsto H(x,p)$ that is $Z^n$ periodic in the first variable $x$, and convex superlinear in the second variable $p$.

For $...

A celebrated theorem in two-dimensional dynamics due to John Franks asserts that every area preserving homeomorphism of the sphere has either two or infinitely many periodic points. In this talk I will describe a new Floer theoretic proof of this...

Let $I= (f_t)_{t \in [0,1]}$ be an identity isotopy on an oriented surface $M$ and denote by $Cont(I)$ the set of points $z \in M$ whose trajectory $I(z)$ is a contractible loop of $M$. A recent result of Olivier Jaulent asserts that there exists an...

The Euler equations for the dynamics of a fluid domain with a free surface can be formulated as a Hamiltonian system in an infinite dimensional phase space. The Hamiltonian has been derived by Zakharov, and the Hamiltonian vector field evokes the...