Systhmes dynamiqueslDynamical Systems
Quelques remarques SW les actions analytiques des rhseaux des groupes de Lie de rang suphrieur RCsumC.
Touks les actions consid&&s ici sent trnalwiques (rklles). Soit IY un sous-groupe d'indioe fini de SL(,rr. L). On montre. en particulier, la rigidit homotopique (globale) de son action affine usuelle sur le tore T" (I/ 2 I<), ainsi que celle de son action projective usuelle sur la sphkre S" ' (pour II > 4). 0 Acadkmie des Sciences/Elsevier. Paris Some remarks on analytic nctions of lattices of Lie groups qf higher runk
A bridged English Version
Let /q, denote the standard affine action of r, a subgroup of finite index of SL(71, Z), on the torus T". In order to prove that it is locally rigid, it is a natural idea to show that every C1 nearby action p is "Iinearizable". From a result on persistance of fixed points of [IO], and a local linearizability result of [I ] (in the analytic case), we get a local linearization of /I around some fixed point. The main ingredient is then to introduce a Siegel nrighborhonci of the fixed point, which is a maximal in\wrimt open set on which the action is linear. We show that the action on the Siege1 neighborhood is ('" -conjugate to the standard action on a punctured torus. i.e. /Jo restricted to the torus with some rational points removed. Holes may exist. in general, as in the case of Katok-Lewis examples which are constructed by a blowing-up process. But in the torus case, a homological consideration leads to Note p&entt?e par ktienne WHYS.