Mixing via families for measure preserving transformations

Rui Kuang ¹   ; Xiangdong Ye ¹

¹ Department of Mathematics University of Science and Technology of China Hefei, Anhui, 230026, P.R. China

Colloquium Mathematicum, Tome 110 (2008) no. 1, pp. 151-165

Résumé

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.

Détail

DOI : 10.4064/cm110-1-5

Keywords: topological dynamics theory recurrence properties via furstenberg families established recent years current paper establish corresponding theory ergodicity via families measurable dynamical systems mds family mathcal subsets mathbb mds mathcal several notions ergodicity related mathcal introduced characterized via weak topology induced hilbert space nbsp mathcal convergence ergodic order ldots positive measure cdots varepsilon mathbb bigcap ne prod varepsilon mathcal proved following statements equivalent nbsp delta * convergence ergodic order nbsp strongly mixing nbsp delta * convergence ergodic order nbsp here delta * dual family family difference sets