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Displaying similar documents to “Ergodic properties of square-free numbers”

Spectrum of multidimensional dynamical systems with positive entropy

B. Kamiński, P. Liardet (1994)

Studia Mathematica

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Applying methods of harmonic analysis we give a simple proof of the multidimensional version of the Rokhlin-Sinaǐ theorem which states that a Kolmogorov ℤ d -action on a Lebesgue space has a countable Lebesgue spectrum. At the same time we extend this theorem to ℤ ∞ -actions. Next, using its relative version, we extend to ℤ ∞ -actions some other general results connecting spectrum and entropy.

Bergelson's theorem for weakly mixing C*-dynamical systems

Rocco Duvenhage (2009)

Studia Mathematica

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We study a nonconventional ergodic average for asymptotically abelian weakly mixing C*-dynamical systems, related to a second iteration of Khinchin's recurrence theorem obtained by Bergelson in the measure-theoretic case. A noncommutative recurrence theorem for such systems is obtained as a corollary.

Ergodic properties of group extensions of dynamical systems with discrete spectra

Mieczysław Mentzen (1991)

Studia Mathematica

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Ergodic group extensions of a dynamical system with discrete spectrum are considered. The elements of the centralizer of such a system are described. The main result says that each invariant sub-σ-algebra is determined by a compact subgroup in the centralizer of a normal natural factor.

Conjugacies between ergodic transformations and their inverses

Geoffrey Goodson (2000)

Colloquium Mathematicae

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We study certain symmetries that arise when automorphisms S and T defined on a Lebesgue probability space (X, ℱ, μ) satisfy the equation S T = T - 1 S . In an earlier paper [6] it was shown that this puts certain constraints on the spectrum of T. Here we show that it also forces constraints on the spectrum of S 2 . In particular, S 2 has to have a multiplicity function which only takes even values on the orthogonal complement of the subspace f ∈ L 2 ( X , ℱ , μ ) : f ( T 2 x ) = f ( x ) . For S and T ergodic satisfying this equation further constraints...

Fixed-point free maps of Euclidean spaces

R. Z. Buzyakova, A. Chigogidze (2011)

Fundamenta Mathematicae

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Our main result states that every fixed-point free continuous self-map of ℝⁿ is colorable. This result can be reformulated as follows: A continuous map f: ℝⁿ → ℝⁿ is fixed-point free iff f̃: βℝⁿ → βℝⁿ is fixed-point free. We also obtain a generalization of this fact and present some examples