Génération des cocycles de degré 2 d'une action mesurable stationnaire de Z^d
Résumé
Let (\Omega,{\cal B},m) be a probability space. To any stationary action T of \mathbb{Z}^d is associated a notion of algebraic cocycle of degree \geq 1, the degree 1 corresponding to the usual sense. This notion was studied by Westman (1971) and Feldman and Moore (1977). Another notion, which presents analogies with the differiential calculus, has been studied by A. and S. Katok (1995). We present a method of generation which associates, to any cocycle in this last sense, an algebraic cocycle of the same degree. Moreover, any cohomology class can be generated so. One consequence of this result is that any cocycle of degree \geq d+1 is a coboundary. Another one is that, in the class of integrable cocycles, the algebraic cohomology of degree \leq d is not trivial, whereas it is in the class of measurable cocycles, as soon as the degree is \geq 2. Finally, this method of generation gives cocycles which verify the ergodic theorem corresponding to their degree.