We show that semisimple actions of l.c.s.c. Abelian groups and cocycles with values in such groups can be used to build new examples of semisimple automorphisms (ℤ-actions) which are relatively weakly mixing extensions of irrational rotations.

Let X be a reflexive Banach space and (Ω,,μ) be a probability measure space. Let T: M(μ;X) → M(μ;X) be a linear operator, where M(μ;X) is the space of all X-valued strongly measurable functions on (Ω,,μ). We assume that T is continuous in the sense that if (fₙ) is a sequence in M(μ;X) and $li{m}_{n\to \infty }fₙ=f$ in measure for some f ∈ M(μ;X), then also $li{m}_{n\to \infty }Tfₙ=Tf$ in measure. Then we consider the functional equation f = (T-I)h, where f ∈ M(μ;X) is given. We obtain several conditions for the existence of h ∈ M(μ;X) satisfying...

We study spectral properties of Anzai skew products $T_{\phi }:²\to ²$ defined by $T_{\phi }(z,\omega )=(e^{2\pi i\alpha }z,\phi \left(z\right)\omega )$, where α is irrational and φ: → is a measurable cocycle. Precisely, we deal with the case where φ is piecewise absolutely continuous such that the sum of all jumps of φ equals zero. It is shown that the simple continuous singular spectrum of $T_{\phi }$ on the orthocomplement of the space of functions depending only on the first variable is a “typical” property in the above-mentioned class of cocycles, if α admits a sufficiently fast approximation....

We introduce an invariant of cohomology in Bernoulli shifts, which is used to answer a question about cohomology of Hölder functions with finitary functions whose coding time is integrable. When restricted to the class of Hölder functions, this invariant even provides a criterion of cohomology.