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V. V. Ryzhikov

Publications and source records attributed to V. V. Ryzhikov.

At least 19 recordsLinked to original sources
arXiv2026

The full diapason of convergence rates of Birkhoff averages for ergodic flows

For an ergodic flow, a range of rates of convergence of Birkhoff averages from the maximum rate to an arbitrarily slow rate is realized by choosing the averaging function. For torus windings, the continuity of the averaging functions is ensured. This complements Krengel's classical result on the slow rates of convergence of means for ergodic automorphisms.

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arXiv2020

Entropy invariants of generic actions

We show that the typical dynamical system sometimes begins to behave like a non-deterministic system with a small classical entropy, and this behavior lasts an extremely long time, until the system starts decreasing entropy. Then again it will become almost non-deterministic for a very very long time, but with more smaller classical entropy. Playing on this fact and considering sigma-compact families of measure-preserving zero-entropy transformations, for example, the rectangle exchange transformations, we choose the Kushnirenko entropy so that it is equal to zero for the transformations under consideration, but is infinite for the generic transformation.

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arXiv2020

On measure-preserving rank one transformations

Rank one transformations serve as a source of examples in ergodic theory, showing variety of algebraic, asymptotic and spectral properties of dynamical systems. The properties of a rank one transformation are closely related to the weak closure of its action. In this direction, known and new constructions of transformations and related problems are studied.

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arXiv2019

Spectral properties and approximations of joinings for infinite rank-one actions

An ergodic self-joining of an infinite rank-one transformation is a part of the weak limit of off-diagonal measures. A class of uncountaible cardinality of nonisomorphic transformations with polynomial weak closure is presented. Such actions have minimal self-joinings and some unusual spectral properties. For any set $M$ of positive integers there exists an infinite transformation $T$ such that the products $T\otimes T^m$ have simple singular spectrum as $1<m\in M$, and Lebesgue spectrum as $1<m\notin M$. There are similar effects for the corresponding Gaussian and Poisson suspensions.

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arXiv2019

Weakly homoclinic groups of ergodic actions

This work contains the following results: the trajectory fullness of the homoclinic groups, their connections with factors, K-property, weak multiple mixing; the ergodicity of the weakly homoclinic group for Gauss and Poisson actions; the triviality of the weakly homoclinic group for classes of rank-1 actions. Some open problems are discussed.

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arXiv2014

On spectral multiplicities of Gaussian actions

For any set $M$ of natural numbers there are mixing Gaussian automorphisms and non-mixing Gaussian automorphisms with singular spectrum (as well as some automorphisms which are disjoint from all Gaussian actions) such that $ M\cup\{\infty\}$ is the set of their spectral multiplicities. We show also that for a Gaussian flow $\{G_t\}$ the sets of spectral multiplicities for some automorphisms $G_t$, $t>0$, could be different.

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arXiv2014

Asymmetries in Asymptotic 3-fold Properties of Ergodic Actions

We present: 1) a mixing $Z ^ 2$-action with the following asymmetry of multiple mixing property: for some commuting measure-preserving transformations $S$, $T$ and a sequence $n_j$ $$ \lim_{j\to \infty}μ(A\bigcap S^{-n_j}A\bigcap T^{-n_j}A)=μ(A)^3$$ for all measurable sets $A$, but there is $A_0$, $μ(A_0)=\frac 1 2$, such that $$ \lim_{j\to \infty}μ(A_0\bigcap S^{n_j}A_0\bigcap T^{n_j}A_0)=0;$$ 2) $Z $-actions with the asymmetry of the partial multiple mixing and the partial multiple rigidity: $$ \lim_{j\to \infty}μ(A\bigcap T^{k_j}A\bigcap T^{m_j}A)= \frac23 μ(A)^3+\frac13μ(A),$$ $$ \lim_{j\to \infty}μ(A\bigcap T^{-k_j}A\bigcap T^{-m_j}A)= μ(A)^2;$$ 3) infinite transformations $T$ such that for all $A$, $μ(A)<\infty$, $$\lim_{j\to \infty}μ(A\bigcap T^{k_j}A\bigcap T^{m_j}A)= \frac13μ(A)$$ and $$\lim_{j\to \infty}μ(A\bigcap T^{-k_j}A\bigcap T^{-m_j}A)=0.$$

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arXiv2013

Chacon's Type Ergodic Transformations with Unbounded Arithmetic Spacers

The following generalizations of the Chacon map are proposed: instead of classical constant spacer sequence $(0,1,0)$ let a sequence $(0,s_j,0)$ be one with unbounded $s_j$. (We mention also an analogue of the historical Chacon map with spacer sequences in the form $(0,s_j)$.) This narrow class of rank-one transformations may be abundant source of open questions. All such constructions have partial rigidity, but some other properties could be different. For root sequence, $ s_j= [\sqrt{j}]$, (or $ s_j= [\ln{j}]$) the corresponding action is rigid, moreover it possesses all polynomials in its weak closure. In the linear case $s_j={j}$ we get (as well as for the classical Chacon transformation) the property of minimal self-joinings (MSJ). We present some observations about MSJ, mild mixing, partial mixing, $æ$-mixing, absence of factors, triviality of centralizer and spectral primality, state several problems, and mention exponential "self-similar" Chacon transformations and flows on infinite measure spaces.

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arXiv2013

Ergodic bounded constructions, Disjointness and Weak Limits of Powers

In connection with some recent results by J. Bourgain and H. Abdalaoui, M. Lemanczyk, T. De La Rue (ALR) we present a short proof of Bourgain's theorem on Mobius orthogonality property for bounded rank-one constructions. The proof of the fact (due to ALR) that bounded-recurrent constructions with Markov self-similarity have to be flat-recurrent is simplified as well.(v4 in Russian)

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arXiv2012

Several questions and hypotheses concerning the limit polynomials for Chacon transformation

We study the weak closure $L$ of powers $T^k$ of the non-singular Chacon transformation $T$ with 2-cuts. This is still an open question does $L$ contain any Markov operator except an orthogonal projector to the constants $Θ$ and some polynomials $P(T)$? In this paper we calculate a particular set of limit polynomials $P_m(T) = \lim_{n \to \infty} T^{-mh_n}$, where $m$ is a fixed integer number and $h_n = (3^n-1)/2$ are the sequence of heights of towers in a standard rank one representation of the Chacon map. We show that for any $d \ge 2$ the family of limit polynomials contains infinitely many distinct polynomials of degree $d$. We also formulate hyposeses and open questions concerning the sequence $P_m$ and the entire set $L$.

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arXiv2012

Homogeneous spectrum, Disjointness of Convolutions, and Mixing Properties of Dynamical Systems

In connection with Rokhlin's question on an automorphism with a homogeneous nonsimple spectrum, we indicate a class of measure-preserving maps $T$ such that $T\times T$ has a homogeneous spectrum of multiplicity 2. The automorphisms in question satisfy the condition $σ\astσ\perp σ$, where $σ$ is the spectral measure of $T$. We also show that there is a mixing automorphism possessing the above properties and their higher order analogs. This work has been published in the pilot issue of Selected Russian Mathematics, Vol.1 (1999), no.1, 13-24.

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