We completely describe the weak closure of the powers of the Koopman operator associated to Chacons classical automorphism. We show that weak limits of these powers are the ortho-projector to constants and an explicit family of polynomials. As a cons
equence, we answer negatively the question of alpha-weak mixing for Chacons automorphism.
We study the generalizations of Jonathan Kings rank-one theorems (Weak-Closure Theorem and rigidity of factors) to the case of rank-one R-actions (flows) and rank-one Z^n-actions. We prove that these results remain valid in the case of rank-one flows
. In the case of rank-one Z^n actions, where counterexamples have already been given, we prove partial Weak-Closure Theorem and partial rigidity of factors.
