Confined extensions and non-standard dynamical filtrations

We explore various ways in which a factor sigma -algebra B can sit in a dynamical system X := ( X, A , mu, T ), i.e. we study some possible structures of the extension A -> B. We consider the concepts of super -innovations and standardness of extensions, , which are inspired by the theory of filtrations. An important aspect of our work is the introduction of the notion of confined extensions, , which first interested us because they have no super -innovations. We give several examples and study additional properties of confined extensions, including several lifting results. Then, using T, T - 1 transformations, we show our main result: the existence of non-standard extensions. Finally, this result finds an application to the study of dynamical filtrations, i.e. filtrations of the form (Fn)n <= 0 n ) n <= 0 such that each F n is a factor sigma -algebra. We show that there exist non-standard I -cosy dynamical filtrations. .