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1997 | 73 | 1 | 35-65
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Systems of finite rank

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  • Institut de Mathématiques de Luminy, CNRS-UPR 9016, Case 930-163 avenue de Luminy, F-13288 Marseille Cedex 9, France
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  • [ADA] T. M. Adams, Smorodinsky's conjecture, submitted, 1993.
  • [ADA-FRI] T. M. Adams and N. A. Friedman, Staircase mixing, submitted, 1992.
  • [AGE] O. N. Ageev, Dynamical systems with a Lebesgue component of even multiplicity in the spectrum, Mat. Sb. 136 (1988), 307-319 (in Russian).
  • [ANO-KAT] D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obshch. 23 (1970), 3-36 (in Russian).
  • [ARN-ORN-WEI] P. Arnoux, D. S. Ornstein and B. Weiss, Cutting and stacking, interval exchanges and geometrical models, Israel J. Math. 50 (1985), 160-168.
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  • [BOU] J. Bourgain, On the spectral type of Ornstein's class one tranformations, Israel J. Math. 84 (1993), 53-63.
  • [BUŁ-KWI-SIE] W. Bułatek, J. Kwiatkowski and A. Siemaszko, Finite rank transformations and weak closure theorem, preprint, 1995.
  • [CHA1] R. V. Chacon, A geometric construction of measure preserving transformations, in: Proc. Fifth Berkeley Sympos. Math. Statist. Probab., Vol. II, Part 2, Univ. of California Press, 1965, 335-360.
  • [CHA2] R. V. Chacon, Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc. 22 (1969), 559-562.
  • [CHA3] R. V. Chacon, Spectral properties of measure-preserving transformations, in: Functional Analysis, Proc. Sympos. Monterrey, 1969, Academic Press, New York, 1970, 93-107.
  • [CHO-NAD1] J. R. Choksi and M. G. Nadkarni, The maximal spectral type of a rank one transformation, Canad. Math. Bull. 37 (1994), 29-36.
  • [CHO-NAD2] J. R. Choksi and M. G. Nadkarni, The group of eigenvalues of a rank one transformation, ibid. 38 (1995), 42-54.
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  • [FER2] S. Ferenczi, Systèmes de rang un gauche, ibid. 21 (1985), 177-186.
  • [FER3] S. Ferenczi, Tiling and local rank properties of the Morse sequence, Theoret. Comput. Sci. 129 (1994), 369-383.
  • [FER4] S. Ferenczi, Les transformations de Chacon: combinatoire, structure géométrique, lien avec les systèmes de complexité 2n+1, Bull. Soc. Math. France 123 (1995), 271-292.
  • [FER5] S. Ferenczi, Rank and symbolic complexity, Ergodic Theory Dynam. Systems 16 (1996), 663-682.
  • [FER-KWI] S. Ferenczi and J. Kwiatkowski, Rank and spectral multiplicity, Studia Math. 102 (1992), 121-144.
  • [FER-KWI-MAU] S. Ferenczi, J. Kwiatkowski and C. Mauduit, A density theorem for (multiplicity, rank) pairs, J. Analyse Math. 65 (1995), 45-75.
  • [FER-MAU-NOG] S. Ferenczi, C. Mauduit and A. Nogueira, Substitution dynamical systems: algebraic characterization of eigenvalues, Ann. Sci. Ecole Norm. Sup. (4) 29 (1996), 519-533.
  • [FIL-KWI] I. Filipowicz and J. Kwiatkowski, Rank, covering number and simple spectrum, J. Analyse Math., to appear.
  • [FRI-GAB-KIN] N. A. Friedman, P. Gabriel and J. L. King, An invariant for rigid rank-1 transformations, Ergodic Theory Dynam. Systems 8 (1988), 53-72.
  • [FRI-KIN] N. A. Friedman and J. L. King, Rank one lightly mixing, Israel J. Math. 73 (1991), 281-288.
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  • [GOO] G. R. Goodson, The structure of ergodic transformations conjugate to their inverses, preprint, 1995.
  • [GOO-deJ-LEM-RUD] G. R. Goodson, A. del Junco, M. Lemańczyk and D. J. Rudolph, Ergodic transformations conjugate to their inverses by involutions, Ergodic Theory Dynam. Systems 16 (1996), 97-124.
  • [GOO-LEM] G. R. Goodson and M. Lemańczyk, On the rank of a class of bijective substitutions, Studia Math. 96 (1990), 219-230.
  • [HED-MOR] G. A. Hedlund and M. Morse, Symbolic dynamics, Amer. J. Math. 60 (1938), 815-866.
  • [HOS-MEL-PAR] B. Host, J. F. Mela and F. Parreau, Nonsingular transformations and spectral analysis of measures, Bull. Soc. Math. France 119 (1991), 33-90.
  • [IWA1] A. Iwanik, Cyclic approximation of irrational rotations, Proc. Amer. Math. Soc. 121 (1994), 691-695.
  • [IWA2] A. Iwanik, Cyclic approximation of analytic cocycles over irrational rotations, Colloq. Math. 70 (1996), 73-78.
  • [IWA-LAC] A. Iwanik and Y. Lacroix, Some constructions of strictly ergodic non-regular Toeplitz flows, Studia Math. 110 (1994), 191-203.
  • [IWA-SER] A. Iwanik and J. Serafin, Most monothetic extensions are rank-1, Colloq. Math. 66 (1993), 63-76.
  • [deJ1] A. del Junco, Transformations with discrete spectrum are stacking transformations, Canad. J. Math. 24 (1976), 836-839.
  • [deJ2] A. del Junco, A transformation with simple spectrum which is not rank one, ibid. 29 (1977), 655-663.
  • [deJ3] A. del Junco, A simple measure-preserving transformation with trivial centralizer, Pacific J. Math. 79 (1978), 357-362.
  • [deJ4] A. del Junco, A family of counterexamples in ergodic theory, Israel J. Math. 44 (1983), 160-188.
  • [deJ-KEA] A. del Junco and M. S. Keane, On generic points in the Cartesian square of Chacon's transformation, ibid. 5 (1985), 59-69.
  • [deJ-RAH-SWA] A. del Junco, A. M. Rahe and M. Swanson, Chacon's automorphism has minimal self-joinings, J. Analyse Math. 37 (1980), 276-284.
  • [deJ-RUD] A. del Junco and D. J. Rudolph, A rank-one, rigid, simple, prime map, Ergodic Theory Dynam. Systems 7 (1987), 229-247.
  • [KAL] S. Kalikow, Twofold mixing implies threefold mixing for rank-1 transformations, ibid. 4 (1984), 237-259.
  • [KAT-SAT] A. B. Katok and E. A. Sataev, Standardness of automorphisms of transposition of intervals and flows on surfaces, Mat. Zametki 20 (1976), 479-488 (in Russian); English transl.: Math. Notes (1977), 826-831.
  • [KAT-STE] A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspekhi Mat. Nauk 22 (5) (1967), 81-106 (in Russian); English transl.: Russian Math. Surveys 22 (5) (1967), 63-75.
  • [KEA1] M. S. Keane, Generalized Morse sequences, Z. Wahrsch. Verw. Gebiete 10 (1968), 335-353.
  • [KEA2] M. S. Keane, Interval exchange transformations, Math. Z. 141 (1975), 25-31.
  • [KEY-NEW] H. Keynes and D. Newton, A 'minimal' non-uniquely ergodic interval exchange transformation, Math. Z. 148 (1976), 101-106.
  • [KIN1] J. L. King, The commutant is the weak closure of the powers, for rank-1 transformations, Ergodic Theory Dynam. Systems 6 (1986), 363-384.
  • [KIN2] J. L. King, For mixing transformations rank(T^k)= k·rank(T), Israel J. Math. 56 (1986), 102-122.
  • [KIN3] J. L. King, Joining-rank and the structure of finite-rank mixing transformations, J. Analyse Math. 51 (1988), 182-227.
  • [KIN-THO] J. L. King and J.-P. Thouvenot, A canonical structure theorem for finite joining-rank maps, ibid. 56 (1991), 211-230.
  • [KLE] I. Klemes, The spectral type of the staircase transformation, Tôhoku Math. J. 48 (1996), 247-258.
  • [KLE-REI] I. Klemes and K. Reinhold, Rank one transformations with singular spectral type, preprint, 1994.
  • [KRI1] W. Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc. 149 (1970), 453-464.
  • [KRI2] W. Krieger, On generators in exhaustive σ-algebras of ergodic measure-preserving transformations, Z. Wahrsch. Verw. Gebiete 20 (1971), 75-82.
  • [KWI-LAC] J. Kwiatkowski and Y. Lacroix, Finite rank transformations and weak closure theorem II, preprint, 1996.
  • [KWI-LEM-RUD] J. Kwiatkowski, M. Lemańczyk and D. J. Rudolph, A class of real cocycles having an analytic coboundary modification, Israel J. Math. 87 (1994), 337-360.
  • [LEM1] M. Lemańczyk, The centralizer of Morse shifts, Ann. Sci. Univ. Clermont-Ferrand II 87 (1985), 43-56.
  • [LEM2] M. Lemańczyk, The rank of Morse dynamical systems, Z. Wahrsch. Verw. Gebiete 70 (1985), 33-48.
  • [LEM3] M. Lemańczyk, Toeplitz $Z_2$-extensions, Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), 1-43.
  • [LEM-MEN] M. Lemańczyk and M. K. Mentzen, On metric properties of substitutions, Compositio Math. 65 (1988), 241-263.
  • [LEM-SIK] M. Lemańczyk and A. Sikorski, A class of not local rank one automorphisms arising from continuous substitutions, Probab. Theory Related Fields 76 (1987), 421-428.
  • [MAT-NAD] J. Mathew and M. G. Nadkarni, A measure preserving transformation whose spectrum has Lebesgue component of multiplicity two, Bull. London Math. Soc. 16 (1984), 402-406.
  • [MEN1] M. Mentzen, Some examples of automorphisms with rank r and simple spectrum, Bull. Polish Acad. Sci. Math. 35 (1987), 417-424.
  • [MEN2] M. Mentzen, Ph.D. Thesis, preprint no 2/89, Nicholas Copernicus University, Toruń, 1989.
  • [MEN3] M. Mentzen, Automorphisms with finite exact uniform rank, Studia Math. 100 (1991), 13-24.
  • [voN] J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math. 33 (1932), 587-642.
  • [ORN1] D. S. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Adv. in Math. 4 (1970), 337-352.
  • [ORN2] D. S. Ornstein, On the root problem in ergodic theory, in: Proc. Sixth Berkeley Sympos. Math. Statist. Probab., Univ. of California Press, 1970, 347-356.
  • [ORN3] D. S. Ornstein, A K-automorphism with no square root and Pinsker's conjecture, Adv. in Math. 10 (1973), 89-102.
  • [ORN-RUD-WEI] D. S. Ornstein, D. J. Rudolph and B. Weiss, Equivalence of measure preserving transformations, Mem. Amer. Math. Soc. 262 (1982).
  • [POL] S. H. Polit, Weakly isomorphic transformations need not be isomorphic, Ph.D. Thesis, Stanford University, 1974.
  • [PRO] E. Prouhet, Mémoire sur quelques relations entre les puissances des nombres, C. R. Acad. Sci. Paris 33 (1851), 31.
  • [QUE] M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Lecture Notes in Math. 1294, Springer, 1987.
  • [RAT1] M. Ratner, Horocycle flows are loosely Bernoulli, Israel J. Math. 31 (1978), 122-131.
  • [RAT2] M. Ratner, Horocycle flows: joinings and rigidity of products, Ann. of Math. 118 (1983), 277-313.
  • [RAT3] M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math. 165 (1990), 229-309.
  • [RAU1] G. Rauzy, Une généralisation des développements en fractions continues, Sém. Delange-Pisot-Poitou, exp. 15 (1976-1977), 16 pp.
  • [RAU2] G. Rauzy, Echanges d'intervalles et transformations induites, Acta Arith. 34 (1979), 315-328.
  • [ROB1] E. A. Robinson, Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), 299-314.
  • [ROB2] E. A. Robinson, Mixing and spectral multiplicity, Ergodic Theory Dynam. Systems 5 (1985), 617-624.
  • [ROS] A. Rosenthal, Les sytèmes de rang fini exact ne sont pas mélangeants, preprint, 1984.
  • [ROT] A. Rothstein, Vershik processes: first steps, Israel J. Math. 36 (1980), 205-223.
  • [RUD] D. J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Analyse Math. 35 (1979), 97-122.
  • [RYZ1] V. V. Ryzhikov, Joinings and multiple mixing of finite rank actions, Funktsional. Anal. i Prilozhen. 27 (2) (1993), 63-78 (in Russian); English transl.: Funct. Anal. Appl. 27 (1993), 128-140.
  • [RYZ2] V. V. Ryzhikov, Multiple mixing and local rank of dynamical systems, Funktsional. Anal. i Prilozhen. 29 (2) (1995), 88-91 (in Russian); English transl.: Funct. Anal. Appl. 29 (1995) 143-145.
  • [VEE] W. Veech, The metric theory of interval exchange transformations I, II, III, Amer. J. Math. 106 (1984), 1331-1421.
  • [VER-LIV] A. M. Vershik and A. N. Livshits, Adic models of ergodic transformations, spectral theory, substitutions, and related topics, in: Representation Theory and Dynamical Systems, Adv. Soviet Math. 9, Amer. Math. Soc., 1992, 185-204.
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