All links below can be found on my webpage.

I gave a talk on tuesday 7th about centralizers of smooth diffeomorphisms of the half line. Here is the abstract of this talk :

"*Let f be a (C ^{∞}) smooth diffeomorphism of the half-line [0,+∞) fixing only the origin, and Z^{r} its centralizer in the group of C^{r} diffeomorphisms. According to well-known results of Szekeres and Kopell, Z^{1} is a one-parameter group. On the other hand, Sergeraert constructed an f whose centralizer Z^{r}, 2 ≤ r ≤ ∞, reduces to the infinite cyclic group generated by f . In this talk, we will show that Z^{r} can actually be a proper dense and uncountable subgroup of Z^{1}.*"

The main result is proved in this article (almost identical to the arxiv version). Since then, I showed that one can actually improve the previous construction so that Z^{r} contains any given Liouville number (see preprint).

Moreover, Christian Bonatti and I recently proved that if one fixes the regularity r<+∞, then, with a more elaborate construction, for *any* irrational number *a* (without any arithmetic condition) one can construct an f whose C^{r} centralizer is exactly Z+aZ. We're currently extending (and hoping to complete) the list of all the possible forms the sequence of centralizers (in every regularity) of a given diffeomorphism can take.

The motivation of these works, related to codimension one foliations on 3-manifold, can be found in my PhD thesis or in this short text I wrote for the Proceedings of the annual Topology Symposium of Japan 2010.