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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 give a negative answer to a question put by Nadkarni: Let S be an ergodic, conservative and nonsingular automorphism on $(X̃{,}_{X̃},m)$. Consider the associated unitary operators on $L²(X̃{,}_{X̃},m)$ given by $U{̃}_{S}f=\surd (d\left(m\circ S\right)/dm)·\left(f\circ S\right)$ and $\phi ·U{̃}_S$, where φ is a cocycle of modulus one. Does spectral isomorphism of these two operators imply that φ is a coboundary? To answer it negatively, we give an example which arises from an infinite measure-preserving transformation with countable Lebesgue spectrum.

We give a positive answer to the problem of existence of smooth weakly mixing but not mixing flows on some surfaces. More precisely, on each compact connected surface whose Euler characteristic is even and negative we construct smooth weakly mixing flows which are disjoint in the sense of Furstenberg from all mixing flows and from all Gaussian flows.

Basic ergodic properties of the ELF class of automorphisms, i.e. of the class of ergodic automorphisms whose weak closure of measures supported on the graphs of iterates of T consists of ergodic self-joinings are investigated. Disjointness of the ELF class with: 2-fold simple automorphisms, interval exchange transformations given by a special type permutations and time-one maps of measurable flows is discussed. All ergodic Poisson suspension automorphisms as well as dynamical systems determined...