# Product ${\mathbb{Z}}^{d}$-actions on a Lebesgue space and their applications

Studia Mathematica (1997)

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

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topFilipowicz, 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

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