MIXING AUTOMORPHISMS OF COMPACT GROUPS AND A THEOREM OF SCHLICKEWEI

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1 MIXING AUTOMORPHISMS OF COMPACT GROUPS AND A THEOREM OF SCHLICKEWEI KLAUS SCHMIDT AND TOM WARD Abstract. We prove that every mixing Z d -action by automorphisms of a compact, connected, abeian group is mixing of a orders. 1. Introduction If α is a mixing automorphism of a compact, abeian group X, then α is Bernoui and hence mixing of a orders ([6], [8]). However, if d > 1, and if α is a mixing Z d -action by automorphismsms of a compact, abeian group X, then α need not be mixing of every order ([5]), and the intricate way in which higher order mixing can break down may be used to construct measurabe isomorphism invariants for α ([3]). In [11] the question was raised whether higher order mixing can fai ony if X is disconnected, and a partia resut in this direction was obtained (the absence of nonmixing shapes for Z d -actions on connected groups). In this paper we answer this question by proving that every mixing Z d -action α by automorphisms of a compact, connected, abeian group is mixing of a orders. Even for commuting tora automorphisms this statement is far from obvious, and its proof depends on a highy nontrivia estimate by H.P. Schickewei ([9]) of the maxima number of soutions (v 1,..., v r ) of equations of the form a 1 v a r v r = 1, subject to certain constraints, where the a i and v i ie in an agebraic number fied K. 2. Mutipe mixing and prime ideas Let (X, S, µ) be a standard (or Lebesgue) probabiity space, d 1, and et T : n T n be a measure preserving Z d -action on (X, S, µ). The action T is mixing of order r (or r-mixing, or mixing on r sets) if, 1991 Mathematics Subject Cassification. 22D40, 28C10, 11D61. The second author gratefuy acknowedges support from NSF grant DMS at the Ohio State University. 1

2 2 KLAUS SCHMIDT AND TOM WARD for a sets B 1,..., B r in S, (1) im n Z d and n n for 1 < r ( r ) µ T n (B ) = =1 r µ(b ). In (1) we may obviousy assume that n 1 = 0. Now assume that X is a compact, abeian group (aways assumed to be metrizabe), S = B X is the Bore fied of X, and that µ = λ X is the normaized Haar measure of X. We write ˆX for the dua group of X, denote by x, χ = χ(x) the vaue at x X of a character χ ˆX, and write ˆη for the automorphism ˆη(χ) = χ η, χ ˆX, of ˆX dua to a continuous automorphism η of X. A homomorphism α : n α n from Z d into the group Aut(X) of continuous automorphisms of X is a Z d -action by automorphisms of X. From (2.1) it is cear that a Z d -action α by automorphisms of a compact, abeian group X is r-mixing if and ony if, for a characters χ 1,..., χ r in ˆX with χ i 1 for some i {1,..., r}, (2) im (χ 1 α n1 ) (χ r α nr ) dλ X = 0. n Z d and n n for 1 < r Again we may assume that n 1 = 0 in (2). The equivaence of (1) and (2) is seen by expanding the indicator functions of the sets B i as Fourier series. Before we discuss the higher order mixing properties of Z d -actions by automorphisms of compact, abeian groups we reca the agebraic description of such actions in [2] and [10]. Let R d = Z[u ±1 1,..., u ±1 be the ring of Laurent poynomias with integra coefficients in the commuting variabes u 1,..., u d. If α is a Z d -action by automorphisms of a compact, abeian group X, then the dua group M = ˆX of X becomes an R d -modue under the R d -action defined by (3) f a = m Z d c f (m)β m (a) =1 d ] for a a M and f = m Z d c f (m)u m R d, where u n = u n 1 1 u n d d for every n = (n 1,..., n d ) Z d, and where β n = α n is the automorphism of M = ˆX dua to α n. In particuar, (4) α n (a) = β n (a) = u n a for a n Z d and a M. Conversey, if M is an R d -modue, and if (5) β M n (a) = u n a for every n Z d and a M, then we obtain a Z d -action (6) α M : n α M n = β M n

3 MIXING AUTOMORPHISMS OF COMPACT GROUPS 3 on the compact, abeian group (7) X M = M dua to the Z d -action β M : n β M n on M. In this notation the r-mixing condition (2.2) is equivaent to the condition that, for a nonzero eements (a 1,..., a r ) M r, (8) u m1 a u mr a r 0 whenever m Z d and m m ies outside some sufficienty arge finite subset of Z d for a 1 < r. If M is an R d -modue, then a prime idea p R d is associated with M if p = {f R d : f a = 0} for some a M, and M is associated with a prime idea p R d if p is the ony prime idea in R d which is associated with M. A nonzero Laurent poynomia f R d is a generaized cycotomic poynomia if there exist m, n Z d and a cycotomic poynomia c in a singe variabe such that n 0 and f = u m c(u n ). The foowing theorem was proved in [10]. Theorem 2.1. Let α be a Z d -action by automorphisms of a compact, abeian group X, and et M = ˆX be the R d -modue arising from α via (2.3) (2.4). The foowing conditions are equivaent. (1) α is mixing (i.e. 2-mixing); (2) α m is ergodic for every 0 m Z d ; (3) None of the prime ideas associated with M contains a generaized cycotomic poynomia. If the Z d -action α in Theorem 2.1 is mixing, then the higher order mixing behaviour of α is again determined by the prime ideas associated with M = ˆX. Theorem 2.2. Let α be a Z d -action by automorphisms of a compact, abeian group X, and et M = ˆX be the R d -modue arising from α via (2.3) (2.4). The foowing conditions are equivaent for every r 2. (1) α is r-mixing; (2) For every prime idea p R d associated with M, the Z d -action α R d/p defined in (2.5) (2.7) is r-mixing. Proof. Suppose that α is r-mixing. If p R d is a prime idea associated with M, then there exists an eement a M such that p = {f R d : f a = 0}, and we set Y = R d a M. Then Y = R d /p and Y = Ŷ = X/Y, where Y = {x X : x, a = 1 for a a Y} is the annihiator of Y. Since Y is invariant under the Z d -action β : n β n = α n dua to α, Y is a cosed, α-invariant subgroup of X, and the Z d -action α Y induced by α on Y is a factor of α and hence r-mixing.

4 4 KLAUS SCHMIDT AND TOM WARD Since the R d -modue arising from α Y is equa to Ŷ = Y = R d /p we concude that α Rd/p must be r-mixing. Conversey, if α is not r-mixing, then (2.8) shows that there exists a nonzero eement (a 1,..., a r ) M r and a sequence (n (m) = (n (m) 1,..., n (m) r ), m 1) in (Z d ) r such that im m n (m) n (m) = for 1 < r and u n(m) 1 a 1 + +u n(m) r a r = 0 for every m 1. There exists a Noetherian submodue N M such that {a 1,..., a r } N, and (2.8) impies that the Z d -action α N, which is a quotient of α, is not r-mixing. Since N is Noetherian, the set of (distinct) prime ideas associated with N is finite and equa to {p 1,..., p m }, say. By Theorem VI.5.3 in [4] there exist submodues W 1,..., W m of N such that N/W i is associated with p i for i = 1,..., m, m i=1 W i = {0}, and i S W i {0} for every subset S {1,..., m}. In particuar, the map a (a + W 1,..., a + W m ) from N into K = m i=1 N/W i is injective, and the dua homomorphism from X = K to N = X N is surjective. Hence α N is a factor of α K, so that α K cannot be r-mixing. By appying (2.8) to the R d -modue K we see that there exists a j {1,..., m} such that α N/W j is not r-mixing. Put V = N/W j, p = p j, and use Lemma 3.4 in [3] to find integers 1 t s and submodues V = N s N 0 = {0} such that, for every k = 1,..., s, N k /N k 1 = Rd /q k for some prime idea p q k R d, q k = p for k = 1,..., t, and q k p for i = t + 1,..., s. We choose Laurent poynomias g k q k p, k = t + 1,..., s, and set g = g t+1 g s. Since α V is not r-mixing, (2.8) impies the existence of a nonzero eement (a 1,..., a r ) V r and a sequence (n (m) = (n (m) 1,..., n (m) r ), m 1) in (Z d ) r such that im m n (m) n (m) = whenever 1 < r, and u n(m) 1 a u n(m) r a r = 0 for every m 1. Put b i = g a i, and note that 0 (b 1,..., b r ) (N t ) r, since g a 0 for every nonzero eement a V. There exists a unique integer p {1,..., t} such that (b 1,..., b r ) (N p ) r (N p 1 ) r, and by setting b i = b i + N p 1 N p /N p 1 = Rd /p we obtain that 0 (b 1,..., b r) (N p /N p 1 ) r = (Rd /p) r and u n(m) 1 b u n(m) r b r = 0 for every m 1, so that α Rd/p is not r-mixing by (2.8). Since the prime idea p is associated with the submodue N M, p is aso associated with M, and the theorem is proved. 3. Schickewei s theorem and mixing Theorem 2.2 shows that a Z d -action α by automorphisms of a compact, abeian group X is mixing of order r 2 if and ony if the

5 MIXING AUTOMORPHISMS OF COMPACT GROUPS 5 Z d -actions α R d/p are r-mixing for a prime ideas p R d associated with the R d -modue M = ˆX defined by α (cf. (2.3) (2.8)). In order to be abe to appy this resut we sha characterize those prime ideas p R d for which α R d/p is r-mixing for every r 2. We identify Z with the set of constant poynomias in R d and note that, for every prime idea p R d, p Z is either equa to pz for some rationa prime p = p(p), or to {0}, in which case we set p(p) = 0. Theorem 3.1. Let d 1, and et p R d be a prime idea such that α R d/p is mixing (cf. Theorem 2.1). (1) If p(p) > 0 then α R d/p is r-mixing for every r 2 if and ony if p = (p(p)) = p(p)r d ; (2) If p(p) = 0 then α R d/p is r-mixing for every r 2. Theorem 3.1 (1) foows from Theorem 3.3 (2) of [Sc2]. We postpone the proof of Theorem 3.1 (2) for the moment and ook instead at some of the consequences of that theorem. If α is a Z d -action by automorphisms of a compact, abeian group X with competey positive entropy, then it is mixing of a orders by Theorem 6.5 and Coroary 6.7 in [7]. If the group X is zero-dimensiona, the reverse impication is aso true. Coroary 3.2. Let α be a Z d -action by automorphisms of a compact, abeian, zero-dimensiona group X. The foowing conditions are equivaent. (1) α has competey positive entropy; (2) α is r-mixing for every r 2. Proof. Since X is zero-dimensiona, every prime idea p associated with the R d -modue M = ˆX arising from α via (2.3) (2.4) contains a nonzero constant, so that p(p) > 0. According to Theorem 6.5 in [7], this impies that α has competey positive entropy if and ony if p = p(p) R d for every prime idea p associated with M, and the equivaence of (1) and (2) foows from Theorem 2.2 and Theorem 3.1 (1). The next coroary shows that the higher order mixing behaviour of Z d -actions by automorphisms of compact, connected, abeian groups is quite different from the zero-dimensiona case, and requires no assumptions concerning entropy. Coroary 3.3. Let d 1, and et α be a mixing Z d -action on a compact, connected, abeian group X. Then α is r-mixing for every r 2.

6 6 KLAUS SCHMIDT AND TOM WARD Proof. The group X is connected if and ony if the dua group ˆX is torsion-free, i.e. if and ony if na 0 whenever 0 a ˆX and 0 n Z. We write M = ˆX for the R d -modue defined by α via (2.3) (2.4), note that the connectedness of X impies that p(p) = 0 for every prime idea p R d associated with M, and appy Theorems 2.2 and 3.1 (2). Coroary 3.4. Let A 1,..., A d be commuting automorphism of the n- torus T n = R n /Z n with the property that the Z d -action α : (m 1,..., m d ) α (m1,...,m d ) = A m 1 1 A m d d is mixing. Then α is r-mixing for every r 2. The proof of Theorem 3.1 (2) depends on a resut by Schickewei [9]. Let K be an agebraic number fied of degree D, and et P (K) be the set of paces and P (K) the set of infinite (or archimedean) paces of K. For every v P (K), v denotes the associated absoute vaue, normaized so that a v is equa to the standard absoute vaue a if v P (K) and a Q, and p v = p 1 if v ies above the rationa prime p. Let S, P (K) S P (K), be a finite set of cardinaity s. An eement a K is an S-unit if a v = 1 for every v P (K) S. Theorem 3.5. (Schickewei) Let a 1,..., a n be nonzero eements of K. Then the equation (9) a 1 v a n v n = 1 has not more than (4sD!) 236nD! s 6 soutions (v 1,..., v n ) in S-units such that no proper subsum a i1 v i1 + + a ik v ik vanishes. Proof. Proof of Theorem 3.1 (2) For every fied F we set F = F {0}. Let Q C be the agebraic cosure of Q, and et V (p) = {c = (c 1,..., c d ) (Q ) d : f(c) = 0 for every f p} and V C (p) = {c = (c 1,..., c d ) (C ) d : f(c) = 0 for every f p}. Suppose that α Rd/p is not r-mixing for some r 3, and that r is the smaest integer with this property. According to (2.8) there exists a nonzero eement (a 1,..., a r ) (R d /p) r and a sequence (n (m) = (n (m) 1,..., n (m) r ), m 1) in (Z d ) r such that im m n (m) n (m) = whenever 1 < r, and u n(m) 1 a u n(m) r a r = 0 for every m 1. For simpicity we assume that n (m) n (n) whenever 1 m < n, and that n (m) 1 = 0 for a m 1. The minimaity of r is easiy seen to impy that a i 0 for i = 1,..., r. Choose f i R d such that a i = f i + p, i = 1,..., r, set, for every c V C (p) and

7 MIXING AUTOMORPHISMS OF COMPACT GROUPS 7 m = (m 1,..., m d ) Z d, c m = c m 1 1 c m d, and note that d (10) f 1 (c) + f 2 (c)c n(m) f r (c)c n(m) r = 0 for a c V C (p) and m 1. If V (p) is finite, then V (p) = V C (p) consists of the orbit of a singe point c = (c 1,..., c d ) under the Gaois group Ga[Q : Q], and the assumption that α R d/p is mixing is equivaent to saying that c m 1 whenever 0 m Z d. The evauation map f f(c), f R d, has kerne p, and may thus be regarded as an injective homomorphism from R d /p into C; in particuar, f 1 (c) f r (c) 0. We denote by K the agebraic number fied Q(c) = Q(c 1,..., c d ) and set S = P (K) {v P (K) : c i v 1 for some i {1,..., d}}. Then S is finite, and Schickewei s Theorem 3.5 impies that the equation f 2(c) f 1 (c) v 2 f r(c) f 1 (c) v r = 1 has ony finitey many soutions (v 2,..., v r ) in S-units such that f i1 (c)v i1 + + f ik (c)v ik 0 whenever 1 < i 1 < < i k r. However, the properties of c and S impy that the vectors (c n(m) 2,..., c n(m) r ), m 1, are a distinct, and that c n(m) i is an S-unit for every i = 2,..., r and m 1. From (10) we concude that, for a but finitey many m 1, one of the subsums f i1 (c)c n(m) i f ik (c)c n(m) i k vanishes. For some choice of 1 < i 1 < < i k r we obtain an infinite set M of positive integers such that f i1 (c)c n(m) i f ik (c)c n(m) i k = 0 for every m M, and this is easiy seen to impy that α Rd/p fais to be k-mixing, where k < r, contrary to the minimaity of r. A moment s refection shows that we have now proved enough to obtain Coroary 3.4. For Theorem 3.1 (2) and Coroary 3.3, however, we have to dea with the case where V (p) is infinite. Since p(p) = 0, the natura homomorphism ι : N = R d /p N = Q Z N, defined by a 1 a for every a N, is injective, and we put z i = ι(u i + p) and z d+i = ι(u 1 i +p) for i = 1,..., d. Noether s normaization emma ([1]), appied to the Q-agebra N, aows us to find an integer t {1,..., 2d} and Q-inear functions w 1,..., w t of the eements z 1,..., z 2d such that {w 1,..., w t } is agebraicay independent over Q and each z 1,..., z 2d is integra over Q[w 1,..., w t ]. We choose and fix monic poynomias Q i Q[w 1,..., w t ][y] = Q[w 1,..., w t, y] such that Q i (w 1,..., w t, z i ) = 0 for i = 1,..., 2d and regard each Q i either as a poynomia in y with coefficients in Q[w 1,..., w t ], or as an eement of Q[w 1,..., w t, y]. Put W C (p) = {(c 1,..., c d, c 1 1,..., c 1 d ) : (c 1,..., c d ) V C (p)} C 2d, define a surjective map ω : W C (p) C t by ω(c) = (w 1 (c),..., w t (c))

8 8 KLAUS SCHMIDT AND TOM WARD for every c W C (p), and note that V C (p) = π(w C (p)) (C ) d C d, where π : C 2d C d is the projection onto the first d coordinates. We write R Q for the set of rationa numbers which occur as one of the coefficients of one of the inear maps w i (regarded as a rationa inear map in 2d variabes), or of one of the poynomias Q i (regarded as a poynomia in t + 1 variabes with rationa coefficients), and et P denote a nonempty, finite set of rationa primes which contains every prime divisor appearing in any eement of R (either in the numerator or the denominator). Put K = {a + b 1 : a, b Q} and denote by S P (K) the (finite) set of a paces of K which are either infinite, or which ie above one of the primes in P. There exists an integer D 1 such that, for every β = (β 1,..., β t ) K t and γ = (γ 1,..., γ 2d ) ω 1 (β), the agebraic number fied K(γ) generated by K and (γ 1,..., γ 2d ) has degree (K(γ) : K) D. Then K(γ) has at most D distinct paces above every pace of K, and it foows that the cardinaity S(γ) of the set S(γ) of paces of K(γ) which ie above one of the eements of S is bounded by D S, where S is the cardinaity of S. Let Σ K be the set of S -units, and et β = (β 1,..., β t ) Σ t K t. We caim that every coordinate of every γ = (γ 1,..., γ 2d ) ω 1 (β) is an S(γ)-unit. Indeed, if v P (K) S, and if v P (K(γ)) ies above v, then γ i is a root of the monic poynomia Q i (β, y) K[y], and each coefficient ζ of Q i satisfies that ζ v 1. It foows that γ i v 1 for i = 1,... 2d. In particuar, since γ 1 i = γ i+d for i {1,..., d}, we obtain that γ 1 i v = ( γ i v ) 1 1, so that γ i v = γ i+d v = 1, as caimed. Since Σ is dense in C, the set Ω = π(ω 1 (Σ t )) V (p) is dense in V C (p), and for every c = (c 1,..., c d ) Ω we either have that f 1 (c) = 0 and f 2 (c)c n(m) 2 + +f r (c)c n(m) r = 0 for every m 1, or that f 1 (c) 0, in which case case Schickewei s theorem impies that the equation f 2(c) f 1 (c) v 2 f r(c) f 1 (c) v r = 1 has at most C = (4D S D!) 236(r 1)D! (D S ) 6 distinct soutions (v 2,..., v r ) in S-units for which no subsum f i1 (c)v i1 + + f ik (c)v ik vanishes. For a 1 m < n, k < r, and {i 1,..., i k } {1,..., r} with 1 i 1 < < i k r, we set Φ (m,n) = {c V C (p) : c n(m) i = c n(n) i for i = 2,..., r} and Ψ(i 1,..., i k ) (m) = {c V C (p) : f i1 (c)c n(m) i 1 we have just seen, (11) Ω s m<n C+s+2 {i 1,...,i k } {1,...,r} + + f ik (c)c n(m) i k = 0}. As Ψ(i 1,..., i k ) (m) Φ (m,n)

9 MIXING AUTOMORPHISMS OF COMPACT GROUPS 9 for every s 1. Since the sets appearing in the right hand side of (11) are a cosed subsets of the perfect set V C (p), we obtain that V C (p) = Ψ(i 1,..., i k ) (m) Φ (m,n) s m<n C+s+2 {i 1,...,i k } {1,...,r} for every s 1. As the idea p R d is prime, the variety V C (p) must, for every s 1, be contained in one of the sets Ψ(i 1,..., i k ) (m) or Φ (m,n) with s m < n C + s + 2 and {i 1,..., i k } {1,..., r}. The second possibiity is excuded by our assumption that α Rd/p is mixing, and we concude that there exists, for infinitey many m 1, a subset {i 1,..., i k } {1,..., r} (depending on m) such that V C (p) Ψ(i 1,..., i k ) (m). Since there are ony finitey many such subsets we obtain that α Rd/p fais to be k-mixing for some k < r, contrary to the minimaity of r, exacty as in the case where V (p) is finite. This contradiction impies that α Rd/p is r-mixing for every r 2. References [1] M. Atiyah and I. G. MacDonad: Introduction to Commutative Agebra, Addison Wesey, Reading, Mass. (1969). [2] B. Kitchens and K. Schmidt: Automorphisms of compact groups, Ergod. Th. & Dynam. Sys. 9 (1989), [3] B. Kitchens and K. Schmidt: Mixing Sets and Reative Entropies for Higher Dimensiona Markov Shifts, Preprint (1991). [4] S. Lang: Agebra (2nd Ed.), Addison Wesey, Reading, Mass. (1984). [5] F. Ledrappier: Un champ markovien peut être d entropie nue et méangeant, C. R. Acad. Sci. Paris Ser. A. 287 (1978), [6] D. Lind: The structure of skew products with ergodic group automorphisms, Israe J. Math. 28 (1977), [7] D. Lind, K. Schmidt, and T. Ward: Maher measure and entropy for commuting automorphisms of compact groups, Invent. math. 101 (1990), [8] G. Mies and R.K. Thomas: The breakdown of automorphisms of compact topoogica groups. In: Studies in Probabiity and Ergodic Theory, Advances in Mathematics Suppementary Studies Vo. 2, Academic Press: New York London, 1987, pp [9] H.P. Schickewei: S-unit equations over number fieds, Invent. math. 102 (1990), [10] K. Schmidt: Automorphisms of compact abeian groups and affine varieties, Proc. London Math. Soc. 61 ( 1990), [11] K. Schmidt: Mixing automorphisms of compact groups and a theorem by Kurt Maher, Pacific J. Math. 137 (1989), Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Department of Mathematics, Ohio State University, Coumbus OH 43210, USA