Strange factor representations of type II_1 and pairs of dual dynamical systems

Given a pair of dynamical systems we consider a pair of commuting von Neumann factors of type 11_1. The construction is a generalization of classical von Neumann-Murrey and grouppoid construction. It gives a natural examples of factors with non-unit coupling constant.As the partial cases we obtain Connes-Riffel-Faddeev examples of the representations of rotation algebra. We give also an example for infinite symmetric group.