Random circle diffeomorphisms with periodic orbits are very simple, although the other...

Date
2014/03/28 Fri 16:00 - 17:30
Room
6号館609号室
Speaker
Michele Triestino
Affiliation
ENS Lyon
Abstract

It is possible to define probability measures on the group of C^1 (orientation preserving) circle diffeomorphisms in several ways. Some are nicer: by the end of the 80s, Malliavin and Shavgulidze have found a family of probability measures which we can consider to be Haar-like (though Diff1_+(S^1) is a non-locally compact group). These measures have enjoyed the attention from the world of representation theory.

We begin here a trip through their dynamical side, that is, we study generic properties of circle diffeomorphisms (with respect to the measure). We will explain that a diffeomorphism can only have finitely many periodic orbits (almost surely). Using the classical theory of Sternberg and Kopell, we deduce that the C^1 centralizer of a diffeomorphism with periodic orbits is a.s. trivial.

However, even if we understand the << stable >> set of diffeomorphisms with periodic orbits pretty well, many open questions arise when looking at the remaining << dark matter >>. E.g. has it got positive measure? How do the measures behave under the renormalization operator? The conjectures will be decorated with numerical pictures.