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Mixing

  • Klaus Schmidt
Part of the Progress in Mathematics book series (PM, volume 128)

Abstract

Let d ≥ 1, and let T be a measure preserving ℤ d -action on a probability space (Y, \(\mathfrak{T}\), µ). A non-empty subset F ⊂ ℤ d is mixing for T if, for all collections of sets {Bn: nF} ⊂ \(\mathfrak{T}\),
$$\mathop{{\lim }}\limits_{{k \to \infty }} \mu \left( {\bigcap\limits_{{n \in F}} {{T_{{ - kn}}}({B_n})} } \right) = \coprod\limits_{{n \in F}} {\mu ({B_n})}$$
(27.1)
, and non-mixing otherwise. If T is r-mixing in the sense of (20.9), then every set F ⊂ Z d of cardinality r is mixing, but the reverse implication is far from clear. If F ⊂ ℤ d is a non-empty set then we can translate F and assume that 0F without affecting its mixing behaviour. Furthermore, if F’ ⊂ℤ d is non-mixing, then every set F’F is non-mixing. Finally, if b · F = {bm: mF} ⊂ ℤ d is non-mixing for some positive b ∈ ℚ, then F is non-mixing.

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Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Klaus Schmidt
    • 1
  1. 1.Mathematisches InstitutUniversität WienViennaAustria

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