Mixing via families for measure preserving transformations
Tom 110 / 2008
Streszczenie
In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family (of subsets of \mathbb Z_+) and a MDS (X,\mathcal{B}, \mu, T), several notions of ergodicity related to \mathcal{F} are introduced, and characterized via the weak topology in the induced Hilbert space L^2(\mu).
T is \mathcal{F}-convergence ergodic of order k if for any A_0,\ldots,A_{k} of positive measure, 0=e_0< \cdots< e_k and \varepsilon>0, \{n\in {\mathbb Z}_+:|\mu(\bigcap_{i=0}^k T^{-ne_i}A_i)-\prod_{i=0}^k\mu(A_i)|< \varepsilon\}\in\mathcal{F}. It is proved that the following statements are equivalent: (1) T is \Delta^*-convergence ergodic of order 1; (2) T is strongly mixing; (3) T is \Delta^*-convergence ergodic of order 2. Here \Delta^* is the dual family of the family of difference sets.