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Mixing via families for measure preserving transformations

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Kuang R., Ye X.

Colloquium Mathematicum

2008

tom 110

nr 1

151-165

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,𝓑,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 $A₀,...,A_{k}$ of positive measure, $0 = e₀ < ⋯ < e_{k}$ and ε > 0, ${n ∈ ℤ₊: |μ(⋂_{i=0}^{k} T^{-ne_{i}}A_{i}) - ∏_{i=0}^{k} μ(A_{i})| < ε} ∈ ℱ$. 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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Kuang R.

1 Ye X.

1 2008

Wyniki wyszukiwania

Mixing via families for measure preserving transformations