Mixing via families for measure preserving transformations

Rui Kuang; Xiangdong Ye

Rui Kuang ; Xiangdong Ye

Colloquium Mathematicae, Tome 111 (2008), p. 151-165 / Harvested from The Polish Digital Mathematics Library

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 ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any $A ₀, . . ., A_{k}$ of positive measure, $0 = e ₀ < \dots < e_{k}$ and ε > 0, $n \in ℤ ₊ : | μ (⋂_{i = 0}^{k} T^{- n e_{i}} A_{i}) - \prod_{i = 0}^{k} μ (A_{i}) | < ε \in ℱ$ . It is proved that the following statements are equivalent: (1) T is Δ*-convergence ergodic of order 1; (2) T is strongly mixing; (3) T is Δ*-convergence ergodic of order 2. Here Δ* is the dual family of the family of difference sets.

Publié le : 2008-01-01

Zbl 1142.37007

EUDML-ID : urn:eudml:doc:286097

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     title = {Mixing via families for measure preserving transformations},
     journal = {Colloquium Mathematicae},
     volume = {111},
     year = {2008},
     pages = {151-165},
     zbl = {1142.37007},
     language = {en},
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Rui Kuang; Xiangdong Ye. Mixing via families for measure preserving transformations. Colloquium Mathematicae, Tome 111 (2008) pp. 151-165. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm110-1-5/