Mixing via families for measure preserving transformations
Volume 110 / 2008
Abstract
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 $\mathcal{F}$ (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.
