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Artem Dudko

Publications and source records attributed to Artem Dudko.

At least 19 recordsLinked to original sources
arXiv2026

Classification of indecomposable states on the infinite symmetric inverse semigroup invariant under the infinite symmetric group. Semifinite case

Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $[g_{ij}]$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a finite set. The binary operation in $R_\infty$ is the ordinary matrix multiplication. It is clear that infinite symmetric group $\mathfrak{S}_\infty$ is a subgroup of $R_\infty$. The map $\star:\left[ g_{ij}\right]\mapsto\left[ g_{ji}\right]$ is an involution on $R_\infty$. We call a function $f$ on $R_\infty$ positive definite if for all $r_1, r_2, \ldots, r_n\in R_\infty$ the matrix $\left[ f\left( r_ir_j^\star\right)\right]$ is Hermitian and positive semi-definite. A function $f$ said to be indecomposable if the corresponding $\ast$-representation $π_f$ is a factor-representation. A class of the $\mathfrak{S}_\infty$-invariant functions is defined by the condition $f(rs)=f(sr)$ for all $r\in R_\infty$ and $s\in\mathfrak{S}_\infty$. In this paper we classify all semifinite factor-representations of $R_\infty$ that correspond to the $\mathfrak{S}_\infty$-invariant positive definite functions.

math.RT↗
arXiv2026

Relative invariant subalgebra rigidity for Thompson's group $F$

We prove that Thompson's group $F$ satisfies the relative invariant subalgebra rigidity property with respect to its commutator subgroup: every von Neumann subalgebra of $L(F)$ that is invariant under conjugation by $[F,F]$ is of the form $L(N)$ for some normal subgroup $N \trianglelefteq F$. Along the way, we establish a general factoriality criterion for invariant subalgebras whose hypotheses are met whenever the ambient group is i.c.c., simple, and every faithful ergodic measure-preserving action of it on a probability space is essentially free.

math.OA↗
arXiv2026

Characters and $II_1$-Factor Representations of Full Groups of Cantor Minimal Systems

Let $(X,T)$ be a Cantor minimal system, and let $Γ$ denote either its associated topological full group or the full group of a Bratteli diagram associated with $(X,T)$. In this paper we describe the structure of indecomposable (extreme) characters and the associated $\textrm{II}_1$-factor representations for the group $Γ$ and its commutator subgroup $Γ'$. In particular, we prove that: (1) for every nontrivial indecomposable character $χ$ of $Γ'$, there exists a finite collection (with repetitions allowed) $\{μ_i\}_{i\in I}$ of $T$-invariant ergodic measures on $X$ such that $χ(γ) = \prod_{i\in I} μ_i(Fix(γ))$, for every $γ\in Γ'$, where $Fix(γ) = \{x\in X : γx = x\}$; and (2) each indecomposable character of $Γ$ is the product of an indecomposable character of the form $\prod_{i\in I} μ_i(Fix(γ))$ and a homomorphism from $Γ$ into the unit circle. As a consequence, we show that any finite-type unitary representation of $Γ'$ that does not contain a regular subrepresentation is automatically continuous with respect to the uniform topology on $Γ'$. We also establish a general result on automatic continuity of finite-type unitary representations of infinite groups, which we use in our proofs.

math.GR↗
arXiv2025

Invariant subalgebras rigidity for von Neumann algebras of groups arising as certain semidirect products

We study the ISR (von Neumann invariant subalgebra rigidity) property for certain discrete groups arising as semidirect products from algebraic actions on certain 2-torsion groups, mostly arising as direct products of $\mathbb{Z}_2$. We present, in particular, the first example of an amenable group with the ISR property that admits a non-trivial abelian normal subgroup. Several other examples are discussed, notably including an infinite amenable group whose von Neumann algebra admits precisely one invariant von Neumann subalgebra which does not come from a normal subgroup. We also investigate the form of invariant subalgebras of the group von Neumann algebra of the standard lamplighter group.

math.OA↗
arXiv2025

Measures and dynamics on Pascal-Bratteli diagrams

We introduce and study dynamical systems and measures on stationary generalized Bratteli diagrams $B$ that are represented as the union of countably many classical Pascal-Bratteli diagrams. We describe all ergodic tail invariant measures on $B$. For every probability tail invariant measure $ν_p$ on the classical Pascal-Bratteli diagram, we approximate the support of $ν_p$ by the path space of a subdiagram. By considering various orders on the edges of $B$, we define dynamical systems with various properties. We show that there exist orders such that the sets of infinite maximal and infinite minimal paths are empty. This implies that the corresponding Vershik map is a homeomorphism. We also describe orders on both $B$ and the classical Pascal-Bratteli diagram that generate either uncountably many minimal infinite and uncountably many maximal infinite paths, or uncountably many minimal infinite paths alongside countably infinitely many maximal infinite paths.

math.DS↗
arXiv2025

A character approach to the ISR property

We develop a character approach to study the invariant von Neumann subalgebras rigidity property (abbreviated as the ISR property) introduced in Amrutam-Jiang's work. First, we introduce the non-factorizable regular character property for groups and show that this implies the ISR property for any infinite ICC groups with trivial amenable radical.Various examples are shown to have this property. Second, we apply known classification results on indecomposable characters to show approximately finite groups have the ISR property. Based on this approach, we also construct non-amenable groups with the ISR property while having non-trivial amenable radical or without the non-factorizable regular character property.

math.OA↗
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 $ω\in\{0,1\}^\Bbb N$, an irreducible unitary representation $κ_ω$ of $G$. Every two representations $κ_ω$ and $κ_{ω'}$ are weakly equivalent. They are unitarily equivalent if and only if $ω$ and $ω'$ are tail equivalent. Each $κ_ω$ appears as the Koopman representation associated with some ergodic $G$-quasiinvariant measure (of infinite product type) on the boundary of the tree.

math.DS↗
arXiv2024

Wild attractors for Fibonacci maps

Existence of wild attractors -- attractors whose basin has a positive Lebesgue measure but is not a residual set -- has been one of central themes in one-dimensional dynamics. It has been demonstrated by H. Bruin et al. that Fibonacci maps with a sufficiently flat critical point admit a wild attractor. We propose a constructive trichotomy that describes possible scenarios for the Lebesgue measure of the Fibonacci attractor based on a computable criterion. We use this criterion, together with a computer-assisted proof of existence of a Fibonacci renormalization $2$-cycle for non-integer critical degrees, to demonstrate that Fibonacci maps do not have a wild attractor when the degree of the critical point is $d=3.8$ (and, conjecturally, for $2< d \le 3.8$), and do admit it when $d=5.1$ (and, conjecturally, for $d \ge 5.1$).

math.DS↗
arXiv2023

Characters and IRS's on branch groups and embeddings into hyperfinite factor

Using the construction by Bencs and Tóth of invariant random subgroups on weakly branch groups acting on regular rooted trees we produce uncountably many indecomposable characters on these groups. In fact, we study three types of characters coming from the action of a weakly branch group on a regular tree, paying attention to their similarities and differences. We use obtained results to show that each countable amenable branch group has uncountably many pairwise not quasi-equivalent embeddings into Murray-von Neumann hyperfinite factor. For the canonical character associated with a self-similar group and studied by the second author as a self-similar trace we provide a number of examples when it is explicitly computed.

math.RT↗
arXiv2023

On spectral properties of the Schreier graphs of the Thompson group $F$

In this article we study spectral properties of the family of Schreier graphs associated to the action of the Thompson group $F$ on the interval [0,1]. In particular, we describe spectra of Laplace type operators associated to these Schreier graphs and calculate certain spectral measures associated to the Schreier graph $Υ$ of the orbit of 1/2. As a byproduct we calculate the asymptotics of the return probabilities of the simple random walk on $Υ$ starting at 1/2. In addition, given a Laplace type operator $L$ on a tree-like graph we study relations between the spectral measures of $L$ associated to delta functions of different vertices and the spectrum of $L$.

math.SP↗
arXiv2022

Lower bounds on the Hausdorff dimension of some Julia sets

We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets of some infinitely renormalizable real quadratic polynomials, including the Feigenbaum polynomial $p_{\,\mathrm{Feig}}(z)=z^2+c_{\,\mathrm{Feig}}$. In addition to that, we construct a piecewise constant function on $[-2,2]$ that provides rigorous lower bounds for the Hausdorff dimension of the Julia sets of all quadratic polynomials $p_c(z) = z^2+c$ with $c \in [-2,2]$. Finally, we verify the conjecture of Ludwik Jaksztas and Michel Zinsmeister that the Hausdorff dimension of the Julia set of a quadratic polynomial $p_c(z)=z^2+c$, is a $C^1$-smooth function of the real parameter $c$ on the interval $c\in(c_{\,\mathrm{Feig}},-3/4)$.

math.DS↗
arXiv2020

On spectra of representations and graphs. Erratum

Unfortunately the proof of the main result of [1], Theorem 1, has a flaw. Namely, Lemma 13 used in the proof of Proposition 11 is correct only under an additional assumption that the operator $A$ is normal (adjoint for the one-sided shift operator in $l^2(\mathbb N)$ provides a counterexample). Below we prove a version of Lemma 13 that does not require the normality assumption and apply it to prove Proposition 11. In addition, the same version of the lemma appears in paper [2] (as Lemma 3.1) where it is used in the proof of Theorem 1.6. We also explain here how to use the new version of Lemma 13 to correct the proof of Theorem 1.6 from [2].

math.SP↗
arXiv2019

On Invariant Random Subgroups of Block-Diagonal Limits of Symmetric Groups

We classify the ergodic invariant random subgroups of block-diagonal limits of symmetric groups in the cases when the groups are simple and the associated dimension groups have finite dimensional state spaces. These block-diagonal limits arise as the transformation groups (full groups) of Bratteli diagrams that preserve the cofinality of infinite paths in the diagram. Given a simple full group $G$ admitting only a finite number of ergodic measures on the path-space $X$ of the associated Bratteli digram, we prove that every non-Dirac ergodic invariant random subgroup of $G$ arises as the stabilizer distribution of the diagonal action on $X^n$ for some $n\geq 1$. As a corollary, we establish that every group character $χ$ of $G$ has the form $χ(g) = Prob(g\in K)$, where $K$ is a conjugation-invariant random subgroup of $G$.

math.GR↗
arXiv2019

On the question "Can one hear the shape of a group?" and Hulanicki type theorem for graphs

We study the question of whether it is possible to determine a finitely generated group $G$ up to some notion of equivalence from the spectrum $\mathrm{sp}(G)$ of $G$. We show that the answer is "No" in a strong sense. As the first example we present the collection of amenable 4-generated groups $G_ω$, $ω\in\{0,1,2\}^\mathbb N$, constructed by the second author in 1984. We show that among them there is a continuum of pairwise non-quasi-isometric groups with $\mathrm{sp}(G_ω)=[-\tfrac{1}{2},0]\cup[\tfrac{1}{2},1]$. Moreover, for each of these groups $G_ω$ there is a continuum of covering groups $G$ with the same spectrum. As the second example we construct a continuum of $2$-generated torsion-free step-3 solvable groups with the spectrum $[-1,1]$. In addition, in relation to the above results we prove a version of Hulanicki Theorem about inclusion of spectra for covering graphs.

math.GR↗
arXiv2017

On diagonal actions of branch groups and the corresponding characters

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acts absolutely non-freely on the boundary of the associated rooted tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II$_1$-factor representations, and infinitely many continuous ergodic invariant random subgroups.

math.RT↗