# Connexité des actions lisses de Z^2 sur l'intervalle

*Résumé:* Smooth actions of Z^2 on the segment [0,1], seen as holonomy

representations, describe codimension one foliations on the thick torus

T^2 x [0,1] transverse to the second factor and having the boundary tori

as leaves. As it turns out, studying the space of such actions is

important to understand the space of codimension 1 foliations on ANY

3-manifold.

This motivates the following result, obtained in a joint work with C.

Bonatti: the space of smooth orientation preserving actions of Z^2 on

[0,1], naturally identified to the space of pairs of commuting

increasing diffeomorphisms of [0,1], is connected.

We don't know, however, if it is path-connected. Of course, deforming a

given pair into any other one is not difficult, the space of smooth

increasing diffeomorphisms of the segment being contractible. But we

will see, appealing to some classical results of Szekeres and Kopell,

that the commutativity condition rigidifies the problem immensely. We

will then explain how to take advantage of this rigidity to prove our

result, and why our method doesn't yield path-connectedness.