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Alexandre I. Danilenko

Publications and source records attributed to Alexandre I. Danilenko.

At least 19 recordsLinked to original sources
arXiv2026

Explicit construction of orbit equivalence for rank-one actions

Let $G$ be a discrete countable infinite group. Given two rank-one measure-preserving $G$-actions whose invariant measures are either both finite or both infinite, we give a direct explicit construction, from their cutting-and-stacking parameters, of a Borel orbit equivalence on invariant conull subsets. We also prove a topological counterpart of this assertion, under additional compatibility assumptions, for continuous $(C,F)$-actions of $G$ on non-compact locally compact Cantor spaces.

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arXiv2025

Classification of rank-one actions via the cutting-and-stacking parameters

Let $G$ be a discrete countable infinite group. Let $T$ and $\widetilde T$ be two rank-one $\sigma$-finite measure preserving actions of $G$ and let $\mathcal T$ and $\widetilde {\mathcal T}$ be the cutting-and-stacking parameters that determine $T$ and $\widetilde T$ respectively. We find necessary and sufficient conditions on $\mathcal T$ and $\widetilde{\mathcal T}$ under which $T$ and $\widetilde T$ are isomorphic. We also show that the isomorphism equivalence relation is a $G_\delta$-subset in the Cartesian square of the set of all admissible parameters $\mathcal T$ endowed with the natural Polish topology. If $G$ is amenable and $T$ and $\widetilde T$ are finite measure preserving then we also find necessary and sufficient conditioins on $\mathcal T$ and $\widetilde {\mathcal T}$ under which $\widetilde T$ is a factor of $T$.

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arXiv2025

Irreducible Koopman representations for nonsingular actions on boundaries of rooted trees

Let $G$ be a countable branch group of automorphisms of a spherically homogeneous rooted tree. Under some assumption on finitarity of $G$, we construct, for each sequence $\omega\in\{0,1\}^\Bbb N$, an irreducible unitary representation $\kappa_\omega$ of $G$. Every two representations $\kappa_\omega$ and $\kappa_{\omega'}$ are weakly equivalent. They are unitarily equivalent if and only if $\omega$ and $\omega'$ are tail equivalent. Each $\kappa_\omega$ appears as the Koopman representation associated with some ergodic $G$-quasiinvariant measure (of infinite product type) on the boundary of the tree.

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arXiv2024

Rank-one nonsingular actions of countable groups and their odometer factors

For an arbitrary countable discrete infinite group $G$, nonsingular rank-one actions are introduced. It is shown that the class of nonsingular rank-one actions coincides with the class of nonsingular $(C,F)$-actions. Given a decreasing sequence $\Gamma_1\supsetneq\Gamma_2\supsetneq\cdots$ of cofinite subgroups in $G$ with $\bigcap_{n=1}^\infty\bigcap_{g\in G}g\Gamma_ng^{-1}=\{1_G\}$, the projective limit of the homogeneous $G$-spaces $G/\Gamma_n$ as $n\to\infty$ is a $G$-space. Endowing this $G$-space with an ergodic nonsingular nonatomic measure we obtain a dynamical system which is called a nonsingular odometer. Necessary and sufficient conditions are found for a rank-one nonsingular $G$-action to have a finite factor and a nonsingular odometer factor in terms of the underlying $(C,F)$-parameters. Similar conditions are also found for a rank-one nonsingular $G$-action to be isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact Cantor models are constructed for the nonsingular rank-one extensions of odometers. Several concrete examples are constructed and several facts are proved that illustrate a sharp difference of the nonsingular noncommutative case from the classical finite measure preserving one: odometer actions which are not of rank one, factors of rank-one systems which are not of rank-one, however each probability preserving odometer is a factor of an infinite measure preserving rank-one system, etc.

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arXiv2022

Krieger's type for ergodic nonsingular Poisson actions of non-(T) locally compact groups

It is shown that each non-compact locally compact second countable non-(T) group $G$ possesses non-strongly ergodic weakly mixing IDPFT Poisson actions of arbitrary Krieger's type. These actions are amenable if and only if $G$ is amenable. If $G$ has the Haagerup property then (and only then) these actions can be chosen of 0-type. If $G$ is amenable and unimodular then $G$ has weakly mixing Bernoulli actions of any possible Krieger's type.

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arXiv2022

Explicit rank-one constructions for irrational rotations

For each {\it well approximable} irrational $θ$, we provide an explicit rank-one construction of the $e^{2πiθ}$-rotation $R_θ$ on the circle $\Bbb T$. This solves "almost surely" a problem by del Junco. For {\it every} irrational $θ$, we construct explicitly a rank-one transformation with an eigenvalue $e^{2πiθ}$. For every irrational $θ$, two infinite $σ$-finite invariant measures $μ_θ$ and $μ_θ'$ on $\Bbb T$ are constructed explicitly such that $(\Bbb T,μ_θ, R_θ)$ is {\it rigid} and of rank one and $(\Bbb T,μ_θ', R_θ)$ is of {\it zero type} and of rank one. The centralizer of the latter system consists of just the powers of $R_θ$. Some versions of the aforementioned results are proved under an extra condition on boundedness of the sequence of cuts in the rank-one construction.

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arXiv2021

Haagerup property and Kazhdan pairs via ergodic infinite measure preserving actions

It is shown that a locally compact second countable group $G$ has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free $G$-action $T=(T_g)_{g\in G}$ on an infinite $σ$-finite standard measure space $(X,μ)$ admitting an exhausting $T$-Følner sequence (i.e. a sequence $(A_n)_{n=1}^\infty$ of measured subsets of finite measure such that $A_1\subset A_2\subset\cdots$, $\bigcup_{n=1}^\infty A_n=X$ and $\lim_{n\to\infty}\sup_{g\in K}\frac{μ(T_gA_n\triangle A_n)}{μ(A_n)}= 0$ for each compact $K\subset G$). It is also shown that a pair of groups $H\subset G$ has property (T) if and only if there is a $μ$-preserving $G$-action $S$ on $X$ admitting an $S$-Følner sequence and such that $S\restriction H$ is weakly mixing. These refine some recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint.

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arXiv2021

Krieger's type of nonsingular Poisson suspensions and IDPFT systems

Given an infinite countable discrete amenable group $Γ$, we construct explicitly sharply weak mixing nonsingular Poisson $Γ$-actions of each Krieger's type: $III_λ$, for $λ\in[0,1]$, and $II_\infty$. The result is new even for $Γ=\Bbb Z$. As these Poisson suspension actions are over very special dissipative base, we obtain also new examples of sharply weak mixing nonsingular Bernoulli $Γ$-actions and IDPFT systems of each possible Krieger's type.

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arXiv2020

Nonsingular Poisson Suspensions

The classical Poisson functor associates to every infinite measure preserving dynamical system $(X,μ,T)$ a probability preserving dynamical system $(X^*,μ^*,T_*)$ called the Poisson suspension of $T$. In this paper we generalize this construction: a subgroup Aut$_2(X,μ)$ of $μ$-nonsingular transformations $T$ of $X$ is specified as the largest subgroup for which $T_*$ is $μ^*$-nonsingular. Topological structure of this subgroup is studied. We show that a generic element in Aut$_2(X,μ)$ is ergodic and of Krieger type III$_1$. Let $G$ be a locally compact Polish group and let $A:G\to\text{Aut}_2(X,μ)$ be a $G$-action. We investigate dynamical properties of the Poisson suspension $A_*$ of $A$ in terms of an affine representation of $G$ associated naturally with $A$. It is shown that $G$ has property (T) if and only if each nonsingular Poisson $G$-action admits an absolutely continuous invariant probability. If $G$ does not have property $(T)$ then for each generating probability $κ$ on $G$ and $t>0$, a nonsingular Poisson $G$-action is constructed whose Furstenberg $κ$-entropy is $t$.

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arXiv2020

Ergodic cocycles of IDPFT systems and nonsingular Gaussian actions

It is proved that each Gaussian cocycle over a mildly mixing Gaussian transformation is either a Gaussian coboundary or sharply weak mixing. The class of nonsingular infinite direct products $T$ of transformations $T_n$, $n\in\Bbb N$, of finite type (IDPFT) is studied. It is shown that if $T_n$ is mildly mixing, $n\in\Bbb N$, the sequence of the Radon-Nikodym derivatives of $T_n$ is asymptotically translation quasi-invariant and $T$ is conservative then the Maharam extension of $T$ is sharply weak mixing. This techniques provides a new approach to the nonsingular Gaussian transformations studied recently by Arano, Isono and Marrakchi.

math.DS↗
arXiv2020

Generic nonsingular Poisson suspension is of type $III_1$

It is shown that for a dense $G_δ$-subset of the subgroup of nonsingular transformations (of a standard infinite $σ$-finite measure space) whose Poisson suspensions are nonsingular, the corresponding Poisson suspensions are ergodic and of Krieger's type $III_1$.

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arXiv2019

On the bounded cohomology for ergodic nonsingular actions of amenable groups

Let $Γ$ be an amenable countable discrete group. Fix an ergodic free nonsingular action of $Γ$ on a nonatomic standard probability space. Let $G$ be a compactly generated locally compact second countable group such that the closure of the group of inner automorphisms of $G$ is compact in the natural topology. It is shown that there exists a {\it bounded} ergodic $G$-valued cocycle of $Γ$.

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arXiv2018

Weak mixing for nonsingular Bernoulli actions of countable amenable groups

Let $G$ be an amenable discrete countable infinite group, $A$ a finite set, and $(μ_g)_{g\in G}$ a family of probability measures on $A$ such that $\inf_{g\in G}\min_{a\in A}μ_g(a)>0$. It is shown (among other results) that if the Bernoulli shiftwise action of $G$ on the infinite product space $\bigotimes_{g\in G}(A,μ_g)$ is nonsingular and conservative then it is weakly mixing. This answers in positive a question by Z.~Kosloff who proved recently that the conservative Bernoulli $\Bbb Z^d$-actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for nonsingular actions of $G$.

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arXiv2017

Rank-one actions, their $(C,F)$-models and constructions with bounded parameters

Let $G$ be a discrete countable infinite group. We show that each topological $(C,F)$-action $T$ of $G$ on a locally compact non-compact Cantor set is a free minimal amenable action admitting a unique up to scaling non-zero invariant Radon measure (answer to a question by Kellerhals, Monod and Rørdam). We find necessary and sufficient conditions under which two such actions are topologically conjugate in terms of the underlying $(C,F)$-parameters. If $G$ is linearly ordered Abelian then the topological centralizer of $T$ is trivial. If $G$ is monotileable and amenable, denote by ${\cal A}_G$ the set of all probability preserving actions of $G$ on the unit interval with Lebesgue measure and endow it with the natural topology. We show that the set of $(C,F)$-parameters of all $(C,F)$-actions of $G$ furnished with a suitable topology is a model for ${\cal A}_G$ in the sense of Forman, Rudolph and Weiss. If $T$ is a rank-one transformation with bounded sequences of cuts and spacer maps then we found simple necessary and sufficient conditions on the related $(C,F)$-parameters under which (i) $T$ is rigid, (ii) $T$ is totally ergodic. It is found an alternative proof of Ryzhikov's theorem that if $T$ is totally ergodic and non-rigid rank-one map with bounded parameters then $T$ has MSJ. We also give a more general version of the criterium (by Gao and Hill) for isomorphism and disjointness of two commensurate non-rigid totally ergodic rank-one maps with bounded parameters. It is shown that the rank-one transformations with bounded parameters and no spacers over the last subtowers is a proper subclass of the rank-one transformations with bounded parameters.

math.DS↗
arXiv2017

Infinite measure~preserving~transformations with Radon MSJ

We introduce concepts of Radon MSJ and Radon disjointness for infinite Radon measure preserving homeomorphisms of the locally compact Cantor space. We construct an uncountable family of pairwise Radon disjoint infinite Chacon like transformations. Every such transformation is Radon strictly ergodic, totally ergodic, asymmetric (not isomorphic to its inverse), has Radon MSJ and possesses Radon joinings whose ergodic components are not joinings.

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