2 Feb 2005; Mat-Nat Faculty (UITO), room U1. Wednesday 10:15-12:00

Boris Kruglikov "Vanishing of topological entropy for certain integrable Hamiltonian systems".

I will give now details of the announced theorems:

Topological entropy of quasi-bi-Hamiltonian system is zero. In particular, if the energy w.r.t. potential is big (e.g. the potential vanishes), then the system is represented geometrically by a manifold with a pair of strictly-non-proportional somewhere metrics with the same geodesics. The conclusion is then that all the dynamical characteristics of each of the geodesic flows grows sub-exponentially.

This result implies some non-trivial topological restrictions on the manifolds in question to possess an integrable system of the considered type. On the other hand, it is quite common, as became clear in recent time, for general integrable Hamiltonian system to have positive entropy.

I also show that the result is very accurate in hypotheses. If we allow different type of singularities for Hamiltonian systems or degenerations for metrics, there exist counter-examples.