Product ℤ d -actions on a Lebesgue space and their applications

I. Filipowicz

Studia Mathematica (1997)

  • Volume: 122, Issue: 3, page 289-298
  • ISSN: 0039-3223

Abstract

top
We define a class of ℤ d -actions, d ≥ 2, called product ℤ d -actions. For every such action we find a connection between its spectrum and the spectra of automorphisms generating this action. We prove that for any subset A of the positive integers such that 1 ∈ A there exists a weakly mixing ℤ d -action, d≥2, having A as the set of essential values of its multiplicity function. We also apply this class to construct an ergodic ℤ d -action with Lebesgue component of multiplicity 2 d k , where k is an arbitrary positive integer.

How to cite

top

Filipowicz, I.. "Product $ℤ^d$-actions on a Lebesgue space and their applications." Studia Mathematica 122.3 (1997): 289-298. <http://eudml.org/doc/216376>.

@article{Filipowicz1997,
abstract = {We define a class of $ℤ^d$-actions, d ≥ 2, called product $ℤ^d$-actions. For every such action we find a connection between its spectrum and the spectra of automorphisms generating this action. We prove that for any subset A of the positive integers such that 1 ∈ A there exists a weakly mixing $ℤ^d$-action, d≥2, having A as the set of essential values of its multiplicity function. We also apply this class to construct an ergodic $ℤ^d$-action with Lebesgue component of multiplicity $2^d k$, where k is an arbitrary positive integer.},
author = {Filipowicz, I.},
journal = {Studia Mathematica},
keywords = {$ℤ^d$-action; spectral theorem; spectrum; spectral multiplicity function; weakly mixing -action; ergodic -action},
language = {eng},
number = {3},
pages = {289-298},
title = {Product $ℤ^d$-actions on a Lebesgue space and their applications},
url = {http://eudml.org/doc/216376},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Filipowicz, I.
TI - Product $ℤ^d$-actions on a Lebesgue space and their applications
JO - Studia Mathematica
PY - 1997
VL - 122
IS - 3
SP - 289
EP - 298
AB - We define a class of $ℤ^d$-actions, d ≥ 2, called product $ℤ^d$-actions. For every such action we find a connection between its spectrum and the spectra of automorphisms generating this action. We prove that for any subset A of the positive integers such that 1 ∈ A there exists a weakly mixing $ℤ^d$-action, d≥2, having A as the set of essential values of its multiplicity function. We also apply this class to construct an ergodic $ℤ^d$-action with Lebesgue component of multiplicity $2^d k$, where k is an arbitrary positive integer.
LA - eng
KW - $ℤ^d$-action; spectral theorem; spectrum; spectral multiplicity function; weakly mixing -action; ergodic -action
UR - http://eudml.org/doc/216376
ER -

References

top
  1. [A] O. N. Ageev, Dynamical systems with a Lebesgue component of even multiplicity in the spectrum, Mat. Sb. 136 (1988), 307-319 (in Russian). 
  2. [BL] F. Blanchard and M. Lemańczyk, Measure preserving diffeomorphisms with an arbitrary spectral multiplicity, Topol. Methods Nonlinear Anal. 1 (1993), 257-294. Zbl0793.28012
  3. [C] R. V. Chacon, Approximation and spectral multiplicity, in: Lecture Notes in Math. 160, Springer, 1970, 18-27. 
  4. [CFS] I. P. Cornfeld, S. W. Fomin and Y. G. Sinai, Ergodic Theory, Springer, 1982. 
  5. [GKLL] 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
  6. [KL] J. Kwiatkowski, Jr., and M. Lemańczyk, On the multiplicity function of ergodic group extensions. II, preprint. 
  7. [L] M. Lemańczyk, Toeplitz Z 2 -extensions, Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), 1-43. Zbl0647.28013
  8. [MN] J. Mathew and M. G. Nadkarni, A measure preserving transformation whose spectrum has Lebesgue component of finite multiplicity, Bull. London Math. Soc. 16 (1984), 402-406. Zbl0515.28010
  9. [O] V. I. Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR 168 (1966), 776-779 (in Russian). Zbl0152.33404
  10. [P] W. Parry, Topics in Ergodic Theory, Cambridge Univ. Press, 1981. Zbl0449.28016
  11. [Q] M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, Lecture Notes in Math. 1294, Springer, 1987. Zbl0642.28013
  12. [Ro] E. A. Robinson, Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), 299-314. Zbl0519.28008
  13. [Ru] W. Rudin, Fourier Analysis on Groups, Interscience Publ., New York, 1962. Zbl0107.09603

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.