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Amenable groups, stationary measures and partitions with independent iterates

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Abstract

Suppose a discrete amenable group G acts freely on a probability space (X, \(\mathcal{B}\), μ) and {g i } is any mixing sequence of group elements, that is μ(g −1i A ∩ B) → μ(A)μ(B) for all A, B ∈ \(\mathcal{B}\). Then given any finite partition P and ε > 0 there is a subsequence {h j } of {g i } and a partition P′ differing from P on a set of measure less than ε such that the partitions {gP: g ∈ IP′{h j }} are jointly independent, where IP′{h j } denotes the set

$$\{ e_G \} \cup \{ h_{j_k } h_{j_{k - 1} } \cdots h_{j1} :j_1 < j_2 < \cdots < j_k \} $$

consisting of the identity of G together with all finite products of the {h j } taken with indices in decreasing order.

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Authors and Affiliations

  1. Institute of Mathematics, Hebrew University, 91904, Jerusalem, Israel

    Boris Begun

  2. Department of Mathematics, University of Toronto, Toronto, M5S 2E4, Canada

    Andrés del Junco

Authors
  1. Boris Begun
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  2. Andrés del Junco
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Additional information

The Research was conducted while the first author was a postdoctoral fellow at the University of Toronto. He thanks the University for its hospitality.

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Begun, B., del Junco, A. Amenable groups, stationary measures and partitions with independent iterates. Isr. J. Math. 158, 41–63 (2007). https://doi.org/10.1007/s11856-007-0003-0

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  • DOI: https://doi.org/10.1007/s11856-007-0003-0

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