arXiv subjects

Mykyta I. Vieprik

Publications and source records attributed to Mykyta I. Vieprik.

4 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.

math.DS↗

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 $σ$-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 $G_δ$-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 $T$.

math.DS↗

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 $Γ_1\supsetneqΓ_2\supsetneq\cdots$ of cofinite subgroups in $G$ with $\bigcap_{n=1}^\infty\bigcap_{g\in G}gΓ_ng^{-1}=\{1_G\}$, the projective limit of the homogeneous $G$-spaces $G/Γ_n$ 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.

math.DS↗

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.

math.DS↗