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Non singular transformations and spectral analysis of measures. (English) Zbl 0748.43001
Authors’ abstract: This work explores in depth the interactions existing between harmonic analysis of measures and spectral theory of nonsingular dynamical systems. It focuses on the study of some classes of remarkable subgroups of the circle: eigenvalue groups, groups of quasi-invariance of measures \(\ldots,\) the most natural examples of which are defined by diophantine conditions. The conjunction of these points of view leads to many new results in both theories, including some classical problems in Fourier analysis.
Reviewer: T.Nakazi (Sapporo)

MSC:
43A05 Measures on groups and semigroups, etc.
37A99 Ergodic theory
28D15 General groups of measure-preserving transformations
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References:
[1] AARONSON (J.) . - The eigenvalues of non-singular transformations , Israel J. of Math., t. 45, 1983 , p. 297-312. MR 86c:28041 | Zbl 0554.28014 · Zbl 0554.28014 · doi:10.1007/BF02804014
[2] AARONSON (J.) . - The intrinsic normalizing constants of transformations preserving infinite measures , J. Analyse Math., (to appear). Zbl 0644.28013 · Zbl 0644.28013 · doi:10.1007/BF02792898
[3] AARONSON (J.) and NADKARNI (M.) . - L\infty eigenvalues and L2 spectra of non-singular transformations , Proc. London Math. Soc., t. 3 55, 1988 , p. 538-570. MR 88j:28012 | Zbl 0636.28010 · Zbl 0636.28010 · doi:10.1112/plms/s3-55.3.538
[4] ARBAULT (J.) . - Sur l’ensemble de convergence absolue d’une série trigonométrique , Bull. Soc. Math. France, t. 80 4, 1952 , p. 253-317. Numdam | MR 14,1080d | Zbl 0048.04202 · Zbl 0048.04202 · numdam:BSMF_1952__80__253_0 · eudml:86861
[5] BARY (N.) . - A treatise on trigonometric series , vol 2. - Pergamon Press, Oxford, 1964 . MR 30 #1347 | Zbl 0129.28002 · Zbl 0129.28002
[6] BROWN (G.) and MORAN (W.) . - Products of random variables and Kakutani’s criterion for orthogonality of product measures , J. London Math. Soc., t. 2 10, 1975 , p. 401-405. MR 51 #11632 | Zbl 0311.60016 · Zbl 0311.60016 · doi:10.1112/jlms/s2-10.4.401
[7] BROWN (G.) and MORAN (W.) . - Bernoulli measure algebras , Acta Math., t. 132, 1974 , p. 77-109. MR 54 #5743 | Zbl 0278.43004 · Zbl 0278.43004 · doi:10.1007/BF02392109
[8] BOURBAKI (N.) . - Topologie générale, chap. 9 . - Hermann, Paris, 1974 . · Zbl 0337.54001
[9] CONNES (A.) . - Une classification des facteurs de type III , Ann. Sci. École Norm. Sup., t. 6, 1973 , p. 133-252. Numdam | MR 49 #5865 | Zbl 0274.46050 · Zbl 0274.46050 · numdam:ASENS_1973_4_6_2_133_0 · eudml:81916
[10] CONNES (A.) . - On the Hierarchy of W. Krieger , Illinois J. of Math., t. 19 3, 1975 , p. 428-432. Article | MR 54 #946 | Zbl 0311.28014 · Zbl 0311.28014 · minidml.mathdoc.fr
[11] FURSTENBERG (H.) and WEISS (B.) . - The finite multipliers of infinite ergodic transformations , Lecture Notes in Mathematics, 688, Springer, 1978 , p. 127-132. MR 80b:28023 | Zbl 0385.28009 · Zbl 0385.28009
[12] GRAHAM (C.) and MCGEHEE (O.C.) . - Essays in Commutative Harmonic Analysis . - Springer-Verlag, 1979 . MR 81d:43001 | Zbl 0439.43001 · Zbl 0439.43001
[13] HAMACHI (T.) , OKA (Y.) and OSIKAWA (M.) . - A classification of ergodic non-singular transformation groups , Mem. Fac. Sci. Kyushu Univ., t. A 28, 1974 , p. 113-133. MR 55 #5839 | Zbl 0293.28011 · Zbl 0293.28011 · doi:10.2206/kyushumfs.28.113
[14] HAMACHI (T.) , OKA (Y.) and OSIKAWA (M.) . - Flows associated with ergodic non-singular transformation groups , Publ. RIMS, Kyoto Univ., t. 11, 1975 , p. 31-50. Article | MR 52 #10998 | Zbl 0316.28007 · Zbl 0316.28007 · doi:10.2977/prims/1195191686 · minidml.mathdoc.fr
[15] HELSON (H.) . - Compact groups with ordered duals, VII , J. London Math. Soc., t. 2 20, 1979 , p. 509-515. MR 81m:43013 | Zbl 0475.43002 · Zbl 0475.43002 · doi:10.1112/jlms/s2-20.3.509
[16] HELSON (H.) . - Cocycles on the circle , J. Operator Theory, t. 16, 1986 , p. 189-199. MR 88f:22020 | Zbl 0644.43003 · Zbl 0644.43003
[17] HELSON (H.) and MERRILL (K.) . - Cocycles on the circle II . Operator Theory : Advances and Applications, 28 c. - Birkhäuser, 1988 . MR 89j:22008 | Zbl 0673.43004 · Zbl 0673.43004
[18] HOST (B.) , MÉLA (J.F.) and PARREAU (F.) . - Analyse harmonique des mesures , Astérisque, 135-136. - Société Mathématique de France, Paris, 1986 . MR 88a:43005 | Zbl 0589.43001 · Zbl 0589.43001
[19] ITO (Y.) , KAMAE (T.) and SHIOKAWA . - Point spectrum and Haussdorff dimension . Number Theory and Combinatorics, p. 209-227 (ed. J. Akiyama et al.). - World Scientific Publishing Co., Tokyo, 1985 . MR 87h:28018 | Zbl 0614.28013 · Zbl 0614.28013
[20] KAHANE (J.P) and SALEM (R.) . - Ensembles parfaits et séries trigonométriques . - Hermann, 1963 . MR 28 #3279 | Zbl 0112.29304 · Zbl 0112.29304
[21] KAKUTANI (S.) . - On the equivalence of infinite product measures , Ann. of Math., t. 49, 1948 , p. 214-224. MR 9,340e | Zbl 0030.02303 · Zbl 0030.02303 · doi:10.2307/1969123
[22] KILMER (S.J.) and SAEKI (S.) . - On Riesz Product measures : mutual absolute continuity and singularity , Ann. Inst. Fourier, t. 38 2, 1988 , p. 63-69. Numdam | MR 90a:42006 | Zbl 0633.43001 · Zbl 0633.43001 · doi:10.5802/aif.1135 · numdam:AIF_1988__38_2_63_0 · eudml:74803
[23] KRIEGER (W.) . - On ergodic Flows and the Isomorphism of Factors , Math. Ann., t. 223, 1976 , p. 19-70. MR 54 #3430 | Zbl 0332.46045 · Zbl 0332.46045 · doi:10.1007/BF01360278 · eudml:162874
[24] KURATOWSKI (C.) . - Topologie I . - Warszawa-Lwow, 1933 . · JFM 59.0563.02
[25] LEDRAPPIER (F.) . - Des produits de Riesz comme mesures spectrales , Ann. Inst. Henri Poincaré, t. 6 4, 1970 , p. 335-344. Numdam | MR 45 #5785 | Zbl 0239.43001 · Zbl 0239.43001 · numdam:AIHPB_1970__6_4_335_0 · eudml:76924
[26] LINDAHL (L.) and POULSEN (F.) . - Thin sets in Harmonic Analysis , Lecture Notes. - Marcel Dekker, 1971 . MR 52 #14800 | Zbl 0226.43006 · Zbl 0226.43006
[27] LOSERT (V.) and SCHMIDT (K.) . - A class of probability measures arising from some problems in ergodic theory , in Probability measures on groups, Springer Lecture Notes in Mathematics, 706, 1979 , p. 124-138. MR 81a:28016 | Zbl 0467.60011 · Zbl 0467.60011
[28] MACKEY (G.W.) . - Borel structure in groups and their duals , Trans. Am. Math. Soc., t. 85, 1957 , p. 134-165. MR 19,752b | Zbl 0082.11201 · Zbl 0082.11201 · doi:10.2307/1992966
[29] MANDREKAR (V.) and NADKARNI (M.) . - On ergodic quasi-invariant measures on the circle group , J. Funct. Anal., t. 3, 1969 , p. 157-163. MR 39 #407 | Zbl 0174.31203 · Zbl 0174.31203 · doi:10.1016/0022-1236(69)90037-8
[30] MÉLA (J.F.) . - Groupes de valeurs propres des systèmes dynamiques et sous-groupes saturés du cercle , C.R. Acad. Sci. Paris, Série I Math., t. 296, 1983 , p. 419-422. MR 85a:22017 | Zbl 0547.28011 · Zbl 0547.28011
[31] MÉLA (J.F.) . - Quasi-invariant measures and Fourier properties , Commutative Harmonic Analysis, Am. Math. Soc. Contemporary Mathematics, vol. 91, 1989 , p. 193-215. MR 92j:43001 | Zbl 0695.43002 · Zbl 0695.43002
[32] MOORE (C.C.) and SCHMIDT (K.) . - Coboundaries and homomorphisms for non-singular group actions and a problem of H. Helson , Proc. London Math. Soc, t. 3 40, 1980 , p. 443-475. MR 82a:22007 | Zbl 0428.28014 · Zbl 0428.28014 · doi:10.1112/plms/s3-40.3.443
[33] OSIKAWA (M.) . - Point spectra of non-singular flows , Publ. Inst. Math. Sci. Kyoto, t. 13, 1977 , p. 167-172. Article | MR 56 #12232 | Zbl 0369.28016 · Zbl 0369.28016 · doi:10.2977/prims/1195190104 · minidml.mathdoc.fr
[34] PARREAU (F.) . - Ergodicité et pureté des produits de Riesz , Ann. Inst. Fourier, t. 40 2, 1990 , p. 391-405. Numdam | MR 91g:42009 | Zbl 0687.43001 · Zbl 0687.43001 · doi:10.5802/aif.1218 · numdam:AIF_1990__40_2_391_0 · eudml:74882
[35] PARREAU (F.) . - Measures with real spectra , Inventiones Math., t. 98, 1989 , p. 311-330. MR 91f:43004 | Zbl 0707.43002 · Zbl 0707.43002 · doi:10.1007/BF01388856 · eudml:143732
[36] PARTHASARATHY (K.R.) . - Probability measures on metric spaces . - Academic Press, 1967 . MR 37 #2271 | Zbl 0153.19101 · Zbl 0153.19101
[37] PEYRIÈRE (J.) . - Étude de quelques propriétés des produits de Riesz , Ann. Inst. Fourier, t. 25 2, 1975 , p. 127-169. Numdam | MR 53 #8771 | Zbl 0302.43003 · Zbl 0302.43003 · doi:10.5802/aif.557 · numdam:AIF_1975__25_2_127_0 · eudml:74218
[38] QUEFFÉLEC (M.) . - Substitution Dynamical Systems , Spectral Analysis. - Springer Lecture Notes, 1294, 1987 . MR 89g:54094 | Zbl 0642.28013 · Zbl 0642.28013
[39] RUDIN (W.) . - Fourier Analysis on groups . - Interscience Publishers, 1962 . MR 27 #2808 | Zbl 0107.09603 · Zbl 0107.09603
[40] SCHMIDT (K.) . - Cocycles on ergodic transformation groups . - Macmillan lectures in Math, 1, 1977 . MR 58 #28262 | Zbl 0421.28017 · Zbl 0421.28017
[41] SCHMIDT (K.) . - Spectra of ergodic group actions , Israel J. Math., t. 41, 1982 , p. 151-153. MR 83m:28022 | Zbl 0498.28018 · Zbl 0498.28018 · doi:10.1007/BF02760662
[42] VARADARAJAN (V.S.) . - Geometry of Quantum Theory , vol. II. - Van Nostrand, 1970 . MR 57 #11400 | Zbl 0194.28802 · Zbl 0194.28802
[43] VAROPOULOS (N.) . - Groups of continuous functions in Harmonic analysis , Acta Math., t. 125, 1970 , p. 109-154. MR 43 #7868 | Zbl 0214.38102 · Zbl 0214.38102 · doi:10.1007/BF02392332
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