Inverse limit of M -cocycles and applications

Jan Kwiatkowski

Fundamenta Mathematicae (1998)

  • Volume: 157, Issue: 2-3, page 261-276
  • ISSN: 0016-2736

Abstract

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For any m, 2 ≤ m < ∞, we construct an ergodic dynamical system having spectral multiplicity m and infinite rank. Given r > 1, 0 < b < 1 such that rb > 1 we construct a dynamical system (X, B, μ, T) with simple spectrum such that r(T) = r, F*(T) = b, and C ( T ) / w c l T n : n ∈ ℤ = ∞

How to cite

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Kwiatkowski, Jan. "Inverse limit of M -cocycles and applications." Fundamenta Mathematicae 157.2-3 (1998): 261-276. <http://eudml.org/doc/212291>.

@article{Kwiatkowski1998,
abstract = {For any m, 2 ≤ m < ∞, we construct an ergodic dynamical system having spectral multiplicity m and infinite rank. Given r > 1, 0 < b < 1 such that rb > 1 we construct a dynamical system (X, B, μ, T) with simple spectrum such that r(T) = r, F*(T) = b, and $#C(T)/wcl\{T^n: n ∈ ℤ\} = ∞$},
author = {Kwiatkowski, Jan},
journal = {Fundamenta Mathematicae},
keywords = {multiplicity; rank; compact group extension; Morse cocycle; Mentzen's problem; maximal spectral multiplicity; ergodic automorphism},
language = {eng},
number = {2-3},
pages = {261-276},
title = {Inverse limit of M -cocycles and applications},
url = {http://eudml.org/doc/212291},
volume = {157},
year = {1998},
}

TY - JOUR
AU - Kwiatkowski, Jan
TI - Inverse limit of M -cocycles and applications
JO - Fundamenta Mathematicae
PY - 1998
VL - 157
IS - 2-3
SP - 261
EP - 276
AB - For any m, 2 ≤ m < ∞, we construct an ergodic dynamical system having spectral multiplicity m and infinite rank. Given r > 1, 0 < b < 1 such that rb > 1 we construct a dynamical system (X, B, μ, T) with simple spectrum such that r(T) = r, F*(T) = b, and $#C(T)/wcl{T^n: n ∈ ℤ} = ∞$
LA - eng
KW - multiplicity; rank; compact group extension; Morse cocycle; Mentzen's problem; maximal spectral multiplicity; ergodic automorphism
UR - http://eudml.org/doc/212291
ER -

References

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  1. [Ch] R. V. Chacon, A geometric construction of measure preserving transformations, in: Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Part 2, Univ. of California Press, 1965, 335-360. 
  2. [Fe1] S. Ferenczi, Systèmes localement de rang un, Ann. Inst. H. Poincaré Probab. Statist. 20 (1984), 35-51. Zbl0535.28010
  3. [Fe2] S. Ferenczi, Systems of finite rank, Colloq. Math. 73 (1997), 35-65. Zbl0883.28014
  4. [FeKw] S. Ferenczi and J. Kwiatkowski, Rank and spectral multiplicity, Studia Math. 102 (1992), 121-144. Zbl0809.28013
  5. [FeKwMa] S. Ferenczi, J. Kwiatkowski and C. Mauduit, A density theorem for (multiplicity, rank) pairs, J. Anal. Math. 65 (1995), 45-75. Zbl0833.28010
  6. [FiKw] I. Filipowicz and J. Kwiatkowski, Rank, covering number and simple spectrum, ibid. 66 (1995), 185-215. Zbl0849.28008
  7. [GoKwLeLi] G. R. Goodson, J. Kwiatkowski, M. Lemańczyk and P. Liardet, On the multiplicity function of ergodic group extensions of rotations, Studia Math. 102 (1992), 157-174. Zbl0830.28009
  8. [GoLe] G. R. Goodson and M. Lemańczyk, On the rank of a class of bijective substitutions, ibid. 96 (1990), 219-230. Zbl0711.28007
  9. [dJ] A. del Junco, A transformation with simple spectrum which is not rank one, Canad. J. Math. 29 (1977), 655-663. Zbl0335.28010
  10. [Kin] J. King, The commutant is the weak closure of the powers, for rank 1 transformations, Ergodic Theory Dynam. Systems 6 (1986), 363-385. 
  11. [KwLa1] J. Kwiatkowski and Y. Lacroix, Multiplicity rank pairs, J. Anal. Math., to appear. Zbl0894.28008
  12. [KwLa2] J. Kwiatkowski and Y. Lacroix, Finite rank transformations and weak closure theorem, preprint. Zbl1198.37009
  13. [KwRo] J. Kwiatkowski and T. Rojek, A method of solving a cocycle functional equation and applications, Studia Math. 99 (1991), 69-86. Zbl0734.28016
  14. [KwSi] J. Kwiatkowski and A. Sikorski, Spectral properties of G-symbolic Morse shifts, Bull. Soc. Math. France 115 (1987), 19-33. Zbl0624.28014
  15. [M1] M. Mentzen, Some examples of automorphisms with rank r and simple spectrum, Bull. Polish Acad. Sci. Math. 35 (1987), 417-424. Zbl0675.28006
  16. [M2] M. Mentzen, thesis, preprint no. 2/89, Nicholas Copernicus University, Toruń, 1989. 
  17. [Pa] W. Parry, Compact abelian group extensions of discrete dynamical systems, Z. Wahrsch. Verw. Gebiete 13 (1969), 95-113. Zbl0184.26901
  18. [R1] E. A. Robinson, Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), 299-314. Zbl0519.28008
  19. [R2] E. A. Robinson, Mixing and spectral multiplicity, Ergodic Theory Dynam. Systems 5 (1985), 617-624. Zbl0565.28013
  20. [dlR] T. de la Rue, Rang des systèmes dynamiques Gaussiens, preprint, Rouen, 1996. 

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