Ergodic properties of square-free numbers

Francesco Cellarosi; Jakov G. Sinaj

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

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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 $ℤ^{\infty}$ -actions. Next, using its relative version, we extend to $ℤ^{\infty}$ -actions some other general results connecting spectrum and entropy.

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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 $f \in L^{2} (X, ℱ, μ) : f (T^{2} x) = f (x)$ . For S and T ergodic satisfying this equation further constraints...

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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

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