Rigid Poisson suspensions without roots

Examples of rigid Poisson suspensions without roots are presented. The discrete rational component in spectrum of an ergodic automorphism S prevents some roots from existing. If S is tensorly multiplied by an ergodic automorphism of the space with a sigma-finite measure, discrete spectrum disappears in this product, but the memory of it can remain in the form of the absence of roots. In additional conditions, this effect is inherited by the Poisson suspension over the product.Comment: in Russian languag

Polynomial rigidity and spectrum of Sidon automorphisms

We present spectrally disjoint Sidon automorphisms whose tensor squares are isomorphic to a planar shift. Spectra of such automorphisms do not possess the group property. To check the singularity of spectrum, we use polynomial rigidity of operators associated with Kolmogorov linear determinism. In the class of mixing Gaussian and Poisson suspensions we realize new sets of spectral multiplicities.Comment: in Russian languag

Slow decay of correlations for generic mixing automorphisms

Let $\psi(n)\to +0$ and a square-integrable function $f$ be non-zero, then for the typical mixing automorphism $T$ the set $\{n:\, |(T^nf,f) |>\psi( n)\}$ is infinite. The mildly mixing automorphisms $T$ do not have convergences of non-zero averages $\frac 1 {k_n} \sum_{i=1}^{k_n}T^if (x)$ with the rate of $o\left(\frac 1 {k_n}\right)$.Comment: in Russian languag

Quasi-similarity, entropy and disjointness of ergodic systems

Answering Vershik's question we show that quasi-similarity does not conserve the entropy, proving quasi-similarity of all Bernoulli actions of a countable infinite group. We prove also the following generalization of Pinsker's theorem: the actions with zero Kirillov-Kushnirenko $P$-entropy and the actions with completely positive $P$-entropy are disjoint. Poisson suspensions are used as examples.Comment: in Russian languag

Tensor-simple spectrum of unitary flows

Unitary flows $T_t$ of dynamic origin are proposed such that for every countable subset $Q\subset (0,+\infty)$ the tensor product $\bigotimes_{q\in Q} T_q$ has simple spectrum. This property is generic for flows preserving the sigma-finite measure.Comment: in Russian languag

Generic extensions of ergodic actions

The article considers generic extensions of measure-preserving actions. We prove that the P-entropy of the generic extensions with finite P-entropy is infinite. This is exploited to obtain the result by Austin, Glasner, Thouvenot, and Weiss that the generic extension of an deterministic action is not isomorphic to it. We show also that generic cocycles are recurrent; as well as typical extensions preserve the singularity of the spectrum, partial rigidity, mildly mixing, and mixing. At the same time, the lifting of some algebraic properties under the generic extension may depend on the statistical properties of the base. The typical measurable families of automorphisms are considered as well. The dynamic behavior of such families is a bit unusual. It is characterized by a combination of the dynamic conformism with the dynamic individualism of the representatives of the generic family.Comment: in Russia

On self-similarities of ergodic flows

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