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Valery V. Ryzhikov

Publications and source records attributed to Valery V. Ryzhikov.

At least 19 recordsLinked to original sources
arXiv2026

Around Kalikow's theorem on multiple mixing

Some results in ergodic theory on joinings and multiple mixing and the proof of Steve Kalikow's theorem on 3-fold mixing of 2-fold mixing rank one automorphisms are discussed.

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arXiv2026

Ergodic Theorems, Almost Invariant Sets, Value Distributions of Time Averagings

It is shown how some deviations of ergodic averages can be structured. The deviations tend to zero almost everywhere. They are asymptotically almost invariant with respect to the action due to averaging. In this situation, the question of the distribution of the values of such deviations is meaningful. It turns out that for any free ergodic $Z^d$-action these distributions can be weakly close to any given distribution if we change the scale on the value line. This article shortly provides related proofs of the ergodic theorems of von Neumann, Birkhoff, Wiener, and Rokhlin's lemma for $Z^d$-actions with an invariant measure.

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arXiv2025

Universal Weighted Averaging for Ergodic Flows

This paper studies homothetic and more general weighted averages for flows. Absolutely continuous convolutions of singular weights are considered, thereby strengthening Kozlov-Treshchev's result on nonuniform averages for ergodic flows. The concept of almost mixing, formulated in terms of homothetic weighted average convergences, is proposed. An example of a non-mixing almost mixing flow is given. It is proven that rigid flows are not almost mixing.

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arXiv2025

Slow convergence almost everywhere of ergodic averages

For an ergodic action of the group $Z^n$ on a probability space and a given arbitrarily slowly decreasing to zero sequence, there exists an integrable function such that the standard ergodic time averages for it converge almost everywhere to the spatial average of the function at a rate that is not asymptotically majorized by this sequence. This generalizes the Krengel effect about the absence of universal estimates for the rate of convergence in Birkhoff's ergodic theorem. The proof uses a weakened version of Rokhlin's lemma for ergodic $Z^n$-actions. It ensures the existence of the required sequence of asymptotically almost invariant sets with given measures. A feature of the construction of such a sequence is that the choice of the next almost invariant set depends on the original function and on the choice of the previous invariant sets. A significant deviation of the ergodic averages from the mean of a positive function can be uniformly realized over extremely large time intervals. The deviation can be greater than a positive constant and differ little from it on arbitrary long time intervals. We not only can achieve the specified deviations arbitrarily far, but the sequence of such deviations from the average can be realized as a sequence wanishing arbitrarily slowly.

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arXiv2025

Slow convergence of Birkhoff ergodic averages

Let $\{T^z\}$ be an ergodic action of the group $Z^n$ by automorphisms of the probability space $(X,m)$, $\sum_{i}^\infty a_i<\infty$, $a_i>0$. For any sequence $M_k\to +\infty$ there exist $N_k>M_k$ and a function $ f\in L_1(X,m)$ such that $$m\left(\ x:\ \left|\, A(x,N_k,f) - \int f \, dm\, \right|\ >\ a_k \ \right)\ \to\ 1,$$ where $$A(x,N,f) =\frac 1 {N^n} \sum_{z\in Q_N} f(T^{z}x),$$ $$Q_N=\{(z_1,\dots,z_n)\}\, :\, 1\leq z_1,\dots,z_n\leq N\}.$$

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arXiv2025

Recurrence of integral zeros for ergodic flows

Let a flow $T_t$ preserve an ergodic probability measure $μ$, $\int f\,dμ=0$, and $μ(A)>0$. Then for almost all $x\in A$, for which $f(x)\neq 0$, there is a sequence ${t_k}\to \infty$ such that $T_{t_k}x\in A$ and $\int_0^{t_k} f(T_sx)ds=0$.

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arXiv2024

Around Krygin-Atkinson, Shneiberg theorems, the recurrence with zero integrals

We recall theorems by Krygin, Atkinson, Shneiberg and propose the following assertion. Let $T_t$ be an ergodic flow on $(X,μ)$, let a function $f$ on $X$ have zero mean, and $μ(A)>0$ for $A\subset X$. Then for almost all $x\in A$ with $f(x)\neq 0$ there exists a sequence $t_k\to\infty$ such that $\int_0^{t_k}f(T_sx)ds=0$ and $T_{t_k}x\in A$.

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arXiv2024

Multiple mixing, 75 years of Rokhlin's problem

In 1949, V.A. Rokhlin introduced new invariants for measure-preserving transformations, called k-fold mixing. Does mixing imply k-fold mixing? -- this problem remains open. We recall shortly some results and discuss related problems.

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arXiv2024

Spectra and joint dynamics of Poisson suspensions for rank-one automorphisms

For every natural $n>1$, there is an operator $T$ of dynamical origin such that its tensor power $T^{\otimes n}$ has singular spectrum, and $T^{\otimes (n+1)}$ has absolutely continuous one. For a set $D$ of positive measure there are mixing zero entropy automorphisms $S,T$ such that $S^nD\cap T^nD=\varnothing$ for all $n>0$. The following answers, in particular, Frantzikinakis-Host's question. If $ p(n+1)- p(n), \ \ q(n+1)- q(n)\ \to\ +\infty$, then there is a divergent sequence $ \sum_{n=1}^{N} μ(S^{ p(n)}C\cap T^{ q(n)}C)/N$ for some set $C$ and automorphisms $S,T$ of simple singular spectrum.

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arXiv2024

Non-recurrence and divergent $\bf ({p(n)},{q(n)})$-averages for deterministic automorphisms

We answer the question of Frantzikinakis and Host about the convergence of ergodic $(n^2,n^3)$-averages and consider a more general case. Let sequences ${ p(n)},{ q(n)}$ satisfy the property $ p(n+1)- p(n), \ q(n+1)- q(n)\ \to\ +\infty.$ Then there exist automorphisms $S,T$ with simple singular spectrum and a set $C$ such that the sequence $ \sum_{n=1}^{N} μ(S^{ p(n)}C\cap T^{ q(n)}C)/N$ diverges. We give also example of linear non-recurrence for a pair of mixing suspensions of zero entropy and with singular and Lebesgue parts in their spectra.

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arXiv2024

Non-commuting transformations with non-converging 2-fold ergodic averages

In connection with the results of Tim Austin, and Wen Huang, Song Shao, Xiangdong Ye we present the following assertion: there are infinite automorphisms $S,T$, some set $A$ of positive finite measure and a sequence $N_m$ with $|N_{m+1}|/|N_m|\ \to\infty$ such that $T^iA\cap S^iA=ϕ$ for $i\in [N_{4k}, N_{4k+1}]$ and $T^iA=S^iA$ for $i\in [N_{4k+2}, N_{2k+3}]$. For the corresponding deterministic Gaussian and Poisson suspensions $S_\circ,T_\circ$ over $S,T$ for some $f\in L_\infty$ there is no limit of $ \frac 1 N \sum_{n=1}^N T_\circ^nf\,S_\circ^nf$ in $ L_2$.

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arXiv2024

Typical infinite mixing automorphisms are rank-one

Bashtanov proved that generic mixing automorphisms of probability space with respect to the Alpern-Tikhonov metric had rank one. Using Sidon's constructions, we show that generic infinite mixing automorphisms also are rank-one.

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