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

I. Filipowicz — 1997

Studia Mathematica

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...

Aproximation of Z-cocycles and shift dynamical systems.

I. Filipowicz;  J. Kwiatkowski;  M. Lemanczyk — 1988

Publicacions Matemàtiques

Let Gbar = G{nt, nt | nt+1, t ≥ 0} be a subgroup of all roots of unity generated by exp(2πi/nt}, t ≥ 0, and let τ: (X, β, μ) O be an ergodic transformation with pure point spectrum Gbar. Given a cocycle φ, φ: X → Z2, admitting an approximation with speed 0(1/n1+ε, ε>0) there exists a Morse cocycle φ such that the corresponding transformations τφ and τ...

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