Mixing

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 {B_n: n ∈ F} ⊂ $\mathfrak{T}$,

$$\mathop{{\lim }}\limits_{{k \to \infty }} \mu \left( {\bigcap\limits_{{n \in F}} {{T_{{ - kn}}}({B_n})} } \right) = \coprod\limits_{{n \in F}} {\mu ({B_n})}$$

((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 0 ∈ F 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: m ∈ F} ⊂ ℤ^d is non-mixing for some positive b ∈ ℚ, then F is non-mixing.