Inverse limit of M -cocycles and applications

Jan Kwiatkowski

Fundamenta Mathematicae (1998)

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

Abstract

top
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 =

How to cite

top

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

top
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.