arXiv subjects

V. V. Ryzhikov

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

27 records · Page 2Linked to original sources

arXiv2011

On Disjointness of Mixing Rank One Actions

For flows the rank is an invariant by linear change of time. But what we can say about isomorphisms? It seems that in case of mixing flows this problem is the most difficult. However the known technique of joinings provides non-isomorphism for mixing rank-one flows under linear change of time. For automorphisms we consider another problems (with similar solutions). For example, the staircase cutting-and-stacking construction is set by a height $h_1$ of the first tower and a sequence $\{r_j\}$ of cut numbers. Let us consider two similar constructions: one is set by $(h_1, \{r_j\})$, another is set by ($h_1+1, \{r_j\}$), and $r_j=j$. We prove a general theorem implying the non-isomorphism of these constructions.

math.DS↗

arXiv2011

Local Rank of Ergodic Symmetric $n$-Powers does not exceed $n!n^{-n}$

We prove that local rank of an ergodic symmetric power $T^{\odot n}$ does not exceed $n!n^{-n}$. A. Katok's old results show that this upper bound is exact. We prove also that $T^{\odot n}$ has infinite Rank as $n>1$.

math.DS↗

arXiv2011

On Mixing of Staircase Transformations

We prove the mixing for all staircase transformations satisfied the conditions $r_j/h_j\to 0$ + $r_j\to\infty$.

math.DS↗

arXiv2011

On Mixing Constructions with Algebraic Spacers

We recall a proof of the mixing for almost all Ornstein's stochastic rank one constructions, replace stochastic spacers by special algebraic ones and prove the mixing in this new situation.

math.DS↗

arXiv2011

On Thouvenot's ergodic proof of Roth's theorem

We present a lecture note on Thouvenot's proof of the Roth-Furstenberg theorem and joining proofs of Furstenberg's theorems on multiple progression average mixing for weakly mixing transformations.

math.DS↗

arXiv2011

Weak Closure Theorem for Double Staircase Actions

We introduce a double staircase construction $T$ and show that the weak closure of $\{T^n\}$ is $\{\int, 2^{-m}T^n+(1-2^{-m})\int \ : m\in\N,\ n\in \Z\}.$

math.DS↗

arXiv2011

Simple Spectrum for Tensor Products of Mixing Map Powers

We prove the existence of a mixing rank one map $T$ such that the product $T\otimes T^2\otimes T^3\otimes...$ has simple spectrum. This result is used in S.V. Thikhonov's proof of the existence of mixing transformation with homogeneous spectrum of multiplicity $m>2$.

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arXiv2011

On Mixing Rank One Infinite Transformations

J.-P. Thouvenot and the author showed via different approaches that the centralizer of a mixing rank-one infinite measure preserving transformation was trivial. In this note the author presents his joining proof. We also consider constructions with algebraic spacers as well as a class of Sidon constructions to produce new examples of mixing rank one transformations. In connection with Gordin's question on the existence of homoclinic ergodic actions for a zero entropy system we also discuss Poisson suspensions of some modifications of Sidon rank one constructions.

math.DS↗

arXiv2011

Spectral multiplicity for powers of weakly mixing automorphisms

We study the behavior of maximal multiplicities $mm (R^n)$ for the powers of a weakly mixing automorphism $R$. For some special infinite set $A$ we show the existence of a weakly mixing rank-one automorphism $R$ such that $mm (R^n)=n$ and $mm(R^{n+1}) =1$ for all $n\in A$. Moreover, the cardinality $cardm(R^n)$ of the set of spectral multiplicities for $R^n$ is not bounded. We have $cardm(R^{n+1})=1$ and $cardm(R^n)=2^{m(n)}$, $m(n)\to\infty$, $n\in A$. We also construct another weakly mixing automorphism $R$ with the following properties: $mm(R^{n}) =n$ for $n=1,2,3,..., 2009, 2010$ but $mm(T^{2011}) =1$, all powers $(R^{n})$ have homogeneous spectrum, and the set of limit points of the sequence $\{\frac{mm (R^n)}{n} : n\in \N \}$ is infinite.

math.DS↗