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Mixing via families for measure preserving transformations

Tom 110 / 2008

Rui Kuang, Xiangdong Ye Colloquium Mathematicum 110 (2008), 151-165 MSC: Primary 37A05, 37A25. DOI: 10.4064/cm110-1-5

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.

Autorzy

  • Rui KuangDepartment of Mathematics
    University of Science and
    Technology of China
    Hefei, Anhui, 230026, P.R. China
    e-mail
  • Xiangdong YeDepartment of Mathematics
    University of Science and
    Technology of China
    Hefei, Anhui, 230026, P.R. China
    e-mail

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