A Note on Mixing Properties of Invertible Extensions

The natural invertible extension T̃ of an Nd-action T has been studied by Lacroix. He showed that T̃ may fail to be mixing even if T is mixing for d ≥ 2. We extend this observation by showing that if T is mixing on (k+ 1) sets then T̃ is in general mixing on no more than k sets, simply because Nd has a corner. Several examples are constructed when d = 2: (i) a mixing T for which T̃ (n,m) has an identity factor whenever n ·m < 0; (ii) a mixing T for which T̃ is rigid but T̃ (n,m) is mixing for all (n,m) 6= (0, 0); (iii) a T mixing on 3 sets for which T̃ is not mixing on 3 sets. 1. Invertible Extensions Let T be a measure-preserving N-action on the probability space (X,B, μ). Such an action may be thought of as the natural shift-action on the space { (xn) ∈ X N | xn = T x0 ∀ n ∈ N } ; the projection π0 onto the zero coordinate shows that T is isomorphic to the shift action, so we identify them. The natural invertible extension of T is constructed in [3], and may be thought of as the natural shift action T̃ on X̃ = { (xn) ∈ X Z | xn+m = T xm ∀ m ∈ Z,n ∈ N } . For any sets F ⊂ Z, G ⊂ N let π̃F : X̃ → X , πG : X → X denote the projections. The set X̃ is a probability space with σ-algebra B̃ and measure μ̃ defined as follows. The σ-algebra B̃ is the smallest one containing all sets of the form Am,C = { (xn) ∈ X̃ | xm ∈ C } for m ∈ Z and C ∈ B, and μ̃ is defined via the Daniell-Kolmogorov consistency theorem (see [1, Theorem 1, Chapter IV.6]) from the requirement that μ̃ (Am,C) = Received March 10, 1997. 1980 Mathematics Subject Classification (1991 Revision). Primary 28D15. The authors gratefully acknowledge support from EPSRC award No. 9570016X, N.S.F. grant No. DMS-94-01093, and the hospitality of the Warwick Mathematics Research Institute. 308 G. MORRIS and T. WARD μ(C). Notice that for {m1, . . . ,ms} ⊂ Z and sets C1, . . . , Cs ∈ B, if ` ∈ N has `+ mj ∈ N for all j, then μ̃ ( {(xn) ∈ X̃ | xmj ∈ Cj for j = 1, . . . , s} ) and μ ( T(C1) ∩ · · · ∩ T (Cs) ) coincide. We shall use the following notation: if B̃ ⊂ X̃ is measurable with respect to π̃ Nd (B) then let B = πNd(B̃) ⊂ X. Let T̃+ = T̃ |Nd be the N -action obtained by restricting the invertible extension to N ⊂ Z. The projection π̃Nd : X̃ → XN d realizes T as a factor of T̃+. If the generators of the original N-action are invertible, then π̃Nd is an isomorphism. Definition. The N-action T is mixing on (k + 1) sets if for any A0, A1, . . . , Ak ∈ B, (1) μ ( A0 ∩ T 1A1 ∩ · · · ∩ T kAk ) −→ μ(A0) . . . μ(Ak) as ni →∞, ni−nj →∞ for i 6= j. Here→∞ means leaving finite subsets of N, and ni−nj →∞ means that if ni + ` = nj + m for `,m ∈ N then ` or m→∞. If k = 1 then mixing on (k + 1) sets is called mixing. A Z-action T is said to be mixing on (k+ 1) sets if (1) holds with the vectors nj now allowed to lie in Z. Lacroix [3] has shown, inter alia, that T mixing does not imply that T̃ will be mixing, with an example in which T̃ has an identity factor for some n ∈ Z\N. We extend this by proving the following theorem and illustrating it with several examples in d = 2, including one in which T is mixing but T̃ has an identity factor for every n ∈ Z\ ( N ∪ −N ) . The “corner” 0 ∈ N is distinguished because it must (unlike the Z case) appear in the expression (1) above. This forces the order of mixing to drop. Theorem. If the N-action T is mixing on (k + 1) sets, then the invertible extension T̃ is mixing on k sets. Proof. Assume T is mixing on (k + 1) sets for some k ≥ 1. Let B̃1, . . . , B̃k be sets measurable with respect to π̃ S(N)(B) where S(N) = [−N,N ] d ∩ Z. Write N = (N,N, . . . , N). Let m2(n), . . . ,mk(n) be integer vectors with mi(n) → ∞ and mi(n) −mj(n) → ∞ as n → ∞ for each i 6= j. For each n = 1, 2, . . . let `(n) ∈ N be chosen so that `(n) → ∞, nj(n) = mj(n) + `(n) → ∞ as n → ∞, and nj(n) ∈ N for all n. Notice by construction we have `(n)→∞, nj(n)→∞, `(n)−nj(n)→∞, and for each i 6= j, nj(n) − ni(n) →∞. It follows that if n is large enough to ensure A NOTE ON MIXING PROPERTIES OF INVERTIBLE EXTENSIONS 309 that `(n)−N ∈ N, then we have μ̃ ( B̃1 ∩ T̃ 2B̃2 ∩ · · · ∩ T̃ kB̃k ) = μ̃ ( T̃B̃1 ∩ T̃ B̃2 ∩ · · · ∩ T̃ kB̃k ) = μ̃ ( X̃ ∩ T̃B̃1 ∩ T̃ 2B̃2 ∩ · · · ∩ T̃ kB̃k ) = μ̃ ( X̃ ∩ T̃ ( T̃B̃1 ) ∩ T̃2 ( T̃B̃2 ) ∩ . . . ∩ T̃k ( T̃B̃k )) = μ ( X ∩ TC1 ∩ T 2C2 ∩ · · · ∩ T kCk ) → μ(C1) . . . μ(Ck) = μ̃(T̃B̃1) . . . μ̃(T̃ B̃k) = μ̃(B̃1) . . . μ(B̃k), where Cj = π̃Nd(T̃ B̃j) for each j. It follows that T̃ is mixing on k sets.