Aoki, Nobuo

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AstérisqueNOBUO AOKIErgodic automorphisms of compact metric groupsare isomorphic to Bernoulli shiftsAstérisque, tome 40 (1976), p. 5-10<http://www.numdam.org/item?id=AST_1976__40__5_0>© Société mathématique de France, 1976, tous droits réservés.L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/Publications/Asterisque/) implique l’accord avec les conditions générales d’uti-lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ouimpression systématique est constitutive d’une infraction pénale. Toute copieou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Société Mathématique de France Astérisque 40 (1976) p.5-10 ERGODIC AUTOMORPHISMS OF COMPACT METRIC GROUPS  ARE ISOMORPHIC TO BERNOULLI SHIFTS Nobuo Aoki I wish to discuss its title. Let X be a compact, metric group and y be its normalized Haar measure. Then (X,y) is a Lebesgue space. Let a be an automorphism of X, then a is an inver-tible measure preserving transformation from X onto X . Our problem is concerned with measure theoretic properties of a . Today we will outline a proof of the following Theorem. An ergodic automorphism of a compact3 metric^ abelian group is a Bernoulli shift. In order to outline the Theorem, we prepare the following Lemma 1. Let X be a compact, metric; abelian group and a be an ergodic automorphism of X . Then there exists a sequence of subgroup XpX.p ... ^ X = {e} , invariant with respect to a , such that there are a a-invariant totally disconnected subgroup X^ , a a-invariant infinite dimensional connected subgroup X^n^ and a a-invariant finite dimensional connected subgroup x̂ n̂  , so that a.v(n) is isomorphic to a Bernoulli shift, aiv(n) is a Bernoulli automorphism (see [28]) and a.v(n) is ergodic, and (X/X ,a) is an algebraic factor of (XA(n) x x*n) x x^n) , axaxa) . To show the ergodic system (X̂  ,a) , n > 1, have the 5 N. AOKI Bernoulli properties, let Tr be a compact, connected, abelian group with the character group Q , being the r-vector of rational numbers and a be an automorphism of Tr , then it is sufficient to prove that if (Tr,a) is ergodic then (Tr,a) is Bernoullian. We write Qr = {glfg2r.--} • Let Qn denote a subgroup generated by {ĝ ocr-1 : j ( Z, 1 <_ k <_ n} for n _> 1. Since rank (Qr) = r , there exists an integer m > 0 such that rank (Q ) = rank (Qr) for n > m . We denote by T(Qn) the annihilator of Qn in Tr for n _> 1 . Then we have Lemma 2. (Tr/T(Qn),a) is Bernoullian for n _> 1 . The proof is done by ideas of Katznelson together with the following facts. We may consider a as operating on Rr . So we have the decomposition Rr = v_ a . . . © V © ... © V such that Vj,-p <_ j £ q, are a-invariant, all the eigenvalues of cf|v have modulus 1 , and all the eigenvalues of oi^ ,j f 0 , have the same modulus p. where p < . . . < 1 = p < . . . . < p Lemma 3. If (0) £ QrO V where V = V_ © . . . Q Vc , then R is decomposed by subspaces R (aR = R ) and R with the following properties: there exists an automorphism a* of R and a homomorphisrr X from R into R such that o(x1rX2) = (o9x1i^(X1) + ^X2) / (X!/X2)6R 1 © R 1 / and a|Rrl gives R = v;p © . . . e V0' 6 ERGODIC AUTOMORPHISMS ARE BERNOULLI where V j C V. for j = -p, ... , 0 , and a1 gives R = © ... © V" © ... © V" where V! Q V . for j = -p , ... , q , such that (V" © ... © v" ) n Qr = {0> . Lemma 4. If a acts ergodically on Q , then so is a. rl and Q r n V1 = {0} . ' r-r-. If Q 1 ft (V* © ... © V") ^ {0} , then we again r-r, ^ 1 r decompose R as in Lemma 3. However, since the dimension of R is finite, such the argument ends in finite time. For m _> 0 , we denote by the set of all r-vectors such that each component of a vector consists of an element of {0, +1,... , + m10} . We define V(1,M) = { YM a m £ : a' 6 V(M) , 1 < m < M} ( m=l V(K,K+M) = m=K omaV : aL f V(m) , K < m < K+M} for K > 0 and M > 0 . Then V(1,M) and V(K,K+M) are subsets of R . Lemjna 5. Assume that R is decomposed as follows. If Rr = v_p 0 ... © V , then Vo Ci Qr = {0} , and if Rr = V_p @ . . . © VQ © . . . © Vq , then (V_p e . . . V ) fl Qr = {O-} and (V0 © ... © V ) 1 Qr = {0}. Then there exists K, > 0 such that V(1,M)A V(K,K+M) = {0} for K _> and an odd integer M > 0. In the proofs of Lemmas 3, 4 and 5, we use no results of number theory. We now conclude that (X,a) is Bernoullian. Using theorem and structures of skew product transformations, I can prove the result for the case of non-abelian. But, proofs of these results will appear elsewhere. 7 N. AOKI REFERENCES [l] L.M. Abramov, The entropy of an automorphism of a solenidal group, Theory of Prob. and its Appl. 4 (1959) , 231 - 236. [2] R.L. Adler and B. Weiss, Similarity of automorphisms of the torus, Mem. Amer. Soc. 98 (1970). [ 3 ] N.Aoki and H. Totoki, Ergodic automorphisms of T are Bernoulli transformations, Publ. RIMS, Kyoto Univ. 10 (1975), 535 - 544. [4] D.Z. Arov, The computation of entropy for one class of group endomorphisms, Zap.Meh.Mat.Fak.Kharov, Math. Obshck 30 ( 1 9 6 4 ) , 4 8 - 6 9 . [5] K.R. Berg, Entropy of torus automorphisms, Topological Dynamics (edited by W.H. Gottschalk and F. Hahn), Benjamin New York, (1969), 516 - 519. [6] D. Lind, Ergodic automorphisms of the infinite torus are Bernoulli, Israel J. Math. 10 (1974), 186 - 195. [7] N.A. Friedman and D.S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math. 5 (1971) , 365 - 394. [8] P.R.Halmos and H.Samelson, On monothetic groups, Proc.Nat. Acad.Sci. U.S.A. 28 (1942), 254 - 258. [ 9 ] G. Hochschild, The structure of Lie Groups, Holden-Day, San Francisco, 1965. [ lo] I. Kaplansky, Groups with representations of bounded degree, Canad.J.Math. 1 (1949); 105 - 112. [ll] Y. Katznelson, Ergodic automorphisms of Tn are Bernoulli shifts, Israel J.Math. 10 (1971), 186 - 195. 8 ERGODIC AUTOMORPHISMS ARE BERNOULLI [l2] A.G. Kurosch, The theory of Groups I, Chelsea, New York, 1960. [l3] D.S. Ornstein, Bernoulli shifts with the same entropy are isomorphic. Advances in Math. 4 (1970), 337 - 352. [l4] , Two Bernoulli shifts with infinite entropy are isomorphic, ibid. 5 (1971) , 339 - 348. [l5] , Factors of Bernoulli shifts are Bernoulli shifts, ibid. 5 (1971), 349 - 364. [l6] W. Parry, Intrinsic Markov chains. Trans.Amer. Math.Soc. 112 (1964) , 5 5 - 6 6 . [l7] M.S. Pinsker, Dynamical systems with completely positive and zero entroy, Soviet Mathematics Doklady, 1 (1960), 937 - 938. [l8] L. Pontjagin, Topological Groups, Princeton Univ.Press. Princeton, 1946. [19] V.A. Rohlin, Metric properties of endomorphisms of compact commutative groups, Amer.Math.Soc. Transl. 64 (1967), 244 - 272. [20] , On the fundamental ideas of measure theory. Amer.Math.Soc. Transl. 10 (1962), 1 - 54. [2l] , Exact endomorphisms of a Lebesgue space, Amer. Math.Soc. Transl. 39 (1969), 1 - 36. [22] — , Selected topics from the metric theory of dynamical systems, Amer.Math.Soc. Transl. 49 (1966), 171 - 240. [23] Ya.G. Sinai, Markov partitions and C-diffeomorphisms, Fane.Anal, and its Appl. 2 (1968), 61 - 82. 9 N. AOKI [24] , On the concept of entropy of a dynamical system, Dokl.Akad.Nauk. 124 (1959) , 768 - 771. [25] H. Totoki, Introduction to Ergodic Theory, Kyoritsu Shuppan, Tokyo, 1971 (in Japanese). [26] , Ergodic Theory, Lecture notes series 14, Aarhus Univ. 1969. [27] A. Wiel, L'intégration dans les Groups Topologique et ses Applications, Actualités Sci, Indust. 869, Hermann, Paris, 1940 , 2nd ed. 1951. [28] S.A. Yuzvinskii, Metric properties of endomorphisms of compact groups. Amer.Math.Transi. 66 (1968), 63 - 98. Department of Mathematics Tokyo Metropolitan University Tokyo, Japan 10 

Ergodic automorphisms of compact metric groups are isomorphic to Bernoulli shifts

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Ergodic automorphisms of compact metric groups are isomorphic to Bernoulli shifts

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