arXiv subjects

Sergey Bezuglyi

Publications and source records attributed to Sergey Bezuglyi.

At least 19 recordsLinked to original sources

arXiv2026

Subdiagrams and invariant measures for generalized Bratteli diagrams

The results of this paper contribute to the study of invariant measures of Borel dynamical systems that can be modeled using generalized Bratteli diagrams. In this context, we study tail invariant measures on the path spaces of generalized Bratteli diagrams, allowing countably infinite vertex sets at each level. Our main focus is on subdiagrams of generalized Bratteli diagrams and the problem of extending tail invariant probability measures from vertex and edge subdiagrams to the ambient diagram. We establish necessary and sufficient conditions for the finiteness of such extensions, formulated in terms of incidence matrices and associated stochastic matrices. Several classes of generalized Bratteli diagrams and their subdiagrams are analyzed in detail, including simple, stationary, and bounded size diagrams. We develop constructive, step-by-step procedures for measure extension and for approximating invariant measures by measures supported on suitable subdiagrams. In addition, we provide explicit examples of generalized Bratteli diagrams that admit no probability tail invariant measures, a phenomenon absent for standard Bratteli diagrams with finite vertex sets. Finally, we address convergence questions for sequences of invariant measures arising from approximations by subdiagrams, clarifying the relationship between combinatorial structure and measure-theoretic behavior.

math.DS↗

arXiv2025

Horizontally stationary generalized Bratteli diagrams

Bratteli diagrams with countably infinite levels exhibit a new phenomenon: they can be horizontally stationary. The incidence matrices of these horizontally stationary Bratteli diagrams are infinite banded Toeplitz matrices. In this paper, we study the fundamental properties of horizontally stationary Bratteli diagrams. In these diagrams, we provide an explicit description of ergodic tail invariant probability measures. For a certain class of horizontally stationary Bratteli diagrams, we prove that all ergodic tail invariant probability measures are extensions of measures from odometers. Additionally, we establish conditions for the existence of a continuous Vershik map on the path space of a horizontally stationary Bratteli diagram.

math.DS↗

arXiv2024

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 $\nu_p$ on the classical Pascal-Bratteli diagram, we approximate the support of $\nu_p$ 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↗

arXiv2024

Inverse limit method for generalized Bratteli diagrams and invariant measures

Generalized Bratteli diagrams with a countable set of vertices in every level are models for aperiodic Borel automorphisms. This paper is devoted to the description of all ergodic probability tail invariant measures on the path spaces of generalized Bratteli diagrams. Such measures can be identified with inverse limits of infinite-dimensional simplices associated with levels in generalized Bratteli diagrams. Though this method is general, we apply it to several classes of reducible generalized Bratteli diagrams. In particular, we explicitly describe all ergodic tail invariant probability measures for (i) the infinite Pascal graph and give the formulas for the values of such measures on cylinder sets, (ii) generalized Bratteli diagrams formed by a countable set of odometers, (iii) reducible generalized Bratteli diagrams with uncountable set of ergodic tail invariant probability measures. We also consider the method of measure extension by tail invariance from subdiagrams. We discuss the properties of the Vershik map defined on reducible generalized Bratteli diagrams.

math.DS↗

arXiv2024

Substitution-dynamics and invariant measures for infinite alphabet-path space

We study substitutions on countably infinite alphabet (without compactification) as Borel dynamical systems. We construct stationary and non-stationary generalized Bratteli-Vershik models for a class of such substitutions, known as left determined. In this setting of Borel dynamics, using a stationary generalized Bratteli-Vershik model, we provide a new and canonical construction of shift-invariant measures (both finite and infinite) for the associated class of subshifts.

math.DS↗

arXiv2024

Invariant measures for reducible generalized Bratteli diagrams

In 2010, Bezuglyi, Kwiatkowski, Medynets and Solomyak [Ergodic Theory Dynam. Systems 30 (2010), no.4, 973-1007] found a complete description of the set of probability ergodic tail invariant measures on the path space of a standard (classical) stationary reducible Bratteli diagram. It was shown that every distinguished eigenvalue for the incidence matrix determines a probability ergodic invariant measure. In this paper, we show that this result does not hold for stationary reducible generalized Bratteli diagrams. We consider classes of stationary and non-stationary reducible generalized Bratteli diagrams with infinitely many simple standard subdiagrams, in particular, with infinitely many odometers as subdiagrams. We characterize the sets of all probability ergodic invariant measures for such diagrams and study partial orders under which the diagrams can support a Vershik homeomorphism.

math.DS↗

arXiv2023

Bratteli diagrams in Borel dynamics

Bratteli-Vershik models have been very successfully applied to the study of various dynamical systems, in particular, in Cantor dynamics. In this paper, we study dynamics on the path spaces of generalized Bratteli diagrams that form models for non-compact Borel dynamical systems. Generalized Bratteli diagrams have countably infinite many vertices at each level, thus the corresponding incidence matrices are also countably infinite. We emphasize differences (and similarities) between generalized and classical Bratteli diagrams. Our main results: $(i)$ We utilize Perron-Frobenius theory for countably infinite matrices to establish criteria for the existence and uniqueness of tail-invariant path space measures (both probability and $σ$-finite). $(ii)$ We provide criteria for the topological transitivity of the tail equivalence relation. $(iii)$ We describe classes of stationary generalized Bratteli diagrams (hence Borel dynamical systems) that: $(a)$ do not support a probability tail-invariant measure, $(b)$ are not uniquely ergodic with respect to the tail equivalence relation. $(iv)$ We describe classes of generalized Bratteli diagrams which can or cannot admit a continuous Vershik map and construct a Vershik map which is a minimal homeomorphism of a (non locally compact) Polish space. $(v)$ We provide an application of the theory of stochastic matrices to analyze diagrams with positive recurrent incidence matrices.

math.DS↗

arXiv2023

Measurable multiresolution systems, endomorphisms, and representations of Cuntz relations

The purpose of this paper is to present new classes of function systems as part of multiresolution analyses. Our approach is representation theoretic, and it makes use of generalized multiresolution function systems (MRSs). It further entails new ideas from measurable endomorphisms-dynamics. Our results yield applications that are not amenable to more traditional techniques used on metric spaces. As the main tool in our approach, we make precise new classes of generalized MRSs which arise directly from a dynamical theory approach to the study of surjective endomorphisms on measure spaces. In particular, we give the necessary and sufficient conditions for a family of functions to define generators of Cuntz relations. We find an explicit description of the set of generalized wavelet filters. Our results are motivated in part by analyses of sub-band filters in signal/image processing. But our paper goes further, and it applies to such wider contexts as measurable dynamical systems, and complex dynamics. A unifying theme in our results is a new analysis of endomorphisms in general measure space, and its connection to multi-resolutions, to representation theory, and generalized wavelet systems.

math.DS↗

arXiv2022

IFS measures on generalized Bratteli diagrams

The purpose of the paper is a general analysis of path space measures. Our focus is a certain path space analysis on generalized Bratteli diagrams. We use this in a systematic study of systems of self-similar measures (the term ``IFS measures'' is used in the paper) for both types of such diagrams, discrete and continuous. In special cases, such measures arise in the study of iterated function systems (IFS). In the literature, similarity may be defined by, e.g., systems of affine maps (Sierpinski), or systems of conformal maps (Julia). We study new classes of semi-branching function systems related to stationary Bratteli diagrams. The latter plays a big role in our understanding of new forms of harmonic analysis on fractals. The measures considered here arise in classes of discrete-time, multi-level dynamical systems where similarity is specified between levels. These structures are made precise by prescribed systems of functions which in turn serve to define self-similarity, i.e., the similarity of large scales, and small scales. For path space systems, in our main result, we give a necessary and sufficient condition for the existence of such generalized IFS measures. For the corresponding semi-branching function systems, we further identify the measures which are also shift-invariant.

math.DS↗

arXiv2021

Cohomology of hyperfinite Borel actions

We study cocycles of countable groups $Γ$ of Borel automorphisms of a standard Borel space $(X, \mathcal{B})$ taking values in a locally compact second countable group $G$. We prove that for a hyperfinite group $Γ$ the subgroup of coboundaries is dense in the group of cocycles. We describe all Borel cocycles of the $2$-odometer and show that any such cocycle is cohomologous to a cocycle with values in a countable dense subgroup $H$ of $G$. We also provide a Borel version of Gottschalk-Hedlund theorem.

math.DS↗

arXiv2020

Harmonic analysis invariants for infinite graphs via operators and algorithms

We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral invariants. We focus on particular classes of infinite graphs, including such weighted graphs which arise in electrical network models, as well as new diagrammatic graph representations. We further stress some direct parallels between our present analysis on infinite graphs, on the one hand, and, on the other, specific areas of potential theory, Fourier duality, probability, harmonic functions, sampling/interpolation, and boundary theory. With the use of limit constructions, finite to infinite, and local to global, we outline how our results for infinite graphs may be viewed as extensions of Shannon's theory: Starting with a countable infinite graph $G$, and a suitable fixed positive weight function, we show that there are certain continua (certain ambient sets $X$) extending $G$, and associated notions of interpolation for (Hilbert spaces of) functions on $X$ from their restrictions to the discrete graph $G$.

math.CO↗

arXiv2020

Harmonic analysis on graphs via Bratteli diagrams and path-space measures

The past decade has seen a flourishing of advances in harmonic analysis of graphs. They lie at the crossroads of graph theory and such analytical tools as graph Laplacians, Markov processes and associated boundaries, analysis of path-space, harmonic analysis, dynamics, and tail-invariant measures. Motivated by recent advances for the special case of Bratteli diagrams, our present focus will be on those graph systems $G$ with the property that the sets of vertices $V$ and edges $E$ admit discrete level structures. A choice of discrete levels in turn leads to new and intriguing discrete-time random-walk models. Our main extension (which greatly expands the earlier analysis of Bratteli diagrams) is the case when the levels in the graph system $G$ under consideration are now allowed to be standard measure spaces. Hence, in the measure framework, we must deal with systems of transition probabilities, as opposed to incidence matrices (for the traditional Bratteli diagrams). The paper is divided into two parts, (i) the special case when the levels are countable discrete systems, and (ii) the (non-atomic) measurable category, i.e., when each level is a prescribed measure space with standard Borel structure. The study of the two cases together is motivated in part by recent new results on graph-limits. Our results depend on a new analysis of certain duality systems for operators in Hilbert space; specifically, one dual system of operator for each level. We prove new results in both cases, (i) and (ii); and we further stress both similarities, and differences, between results and techniques involved in the two cases.

math.DS↗

arXiv2020

C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map

We study homeomorphisms of a Cantor set with $k$ ($k < +\infty$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and their certain orbit-cut sub-C*-algebras. In the case that $k\geq 2$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank zero if in addition $(X, σ)$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli-Vershik-Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli-Vershik-Kakutani model) must have at least $k$ vertices at each level, and the image of the index map must consist infinitesimals.

math.OA↗

arXiv2019

Laplace operators in finite energy and dissipation spaces

Recent applications of large network models to machine learning, and to neural network suggest a need for a systematic study of the general correspondence, (i) discrete vs (ii) continuous. Even if the starting point is (i), limit considerations lead to (ii), or, more precisely, to a measure theoretic framework which we make precise. Our motivation derives from graph analysis, e.g., studies of (infinite) electrical networks of resistors, but our focus will be (ii), i.e., the measure theoretic setting. In electrical networks of resistors, one considers pairs (of typically countably infinite), sets $V$ (vertices), $E$ (edges) a suitable subset of $V \times V$, and prescribed positive symmetric functions $c$ on $E$ . A conductance function $c$ is defined on $E$ (edges), or on $V \times V$, but with $E$ as its support. From an initial triple $(V, E, c)$ , one gets graph-Laplacians, generalized Dirichlet spaces (also called energy Hilbert spaces), dipoles, relative reproducing kernel-theory, dissipation spaces, reversible Markov chains, and more. Our main results include: spectral theory and Green's functions for measure theoretic graph-Laplace operators; the theory of reproducing kernel Hilbert spaces related to Laplace operators; a rigorous analysis of the Laplacian on Borel equivalence relations; a new decomposition theory; irreducibility criteria; dynamical systems governed by endomorphisms and measurable fields; orbit equivalence criteria; and path-space measures and induced dissipation Hilbert spaces. We consider several applications of our results to other fields such as machine learning problems, reproducing kernel Hilbert spaces, Gaussian and determinantal processes, and joinings.

math.FA↗

arXiv2018

Markov operators generated by symmetric measures

With view to applications, we here give an explicit correspondence between the following two: (i) the set of symmetric and positive measures $ρ$ on one hand, and (ii) a certain family of generalized Markov transition measures $P$, with their associated Markov random walk models, on the other. By a generalized Markov transition measure we mean a measurable and measure-valued function $P$ on $(V, \mathcal B)$, such that for every $x \in V , P(x; \cdot)$ is a probability measure on $(V, \mathcal B$). Hence, with the use of our correspondence (i) - (ii), we study generalized Markov transitions $P$ and path-space dynamics. Given $P$, we introduce an associated operator, also denoted by $P$ , and we analyze its spectral theoretic properties with reference to a system of precise $L^2$ spaces. Our setting is more general than that of earlier treatments of reversible Markov processes. In a potential theoretic analysis of our processes, we introduce and study an associated energy Hilbert space $\mathcal H_E$, not directly linked to the initial $L^2$-spaces. Its properties are subtle, and our applications include a study of the $P$-harmonic functions. They may be in $\mathcal H_E$, called finite-energy harmonic functions. A second reason for $\mathcal H_E$ is that it plays a key role in our introduction of a generalized Greens function. (The latter stands in relation to our present measure theoretic Laplace operator in a way that parallels more traditional settings of Greens functions from classical potential theory.) A third reason for $\mathcal H_E$ is its use in our analysis of path-space dynamics for generalized Markov transition systems.

math.FA↗

arXiv2018

Graph Laplace and Markov operators on a measure space

The main goal of this paper is to build a measurable analogue to the theory of weighted networks on infinite graphs. Our basic setting is an infinite $σ$-finite measure space $(V, \mathcal B, μ)$ and a symmetric measure $ρ$ on $(V\times V, \mathcal B\times \mathcal B)$ supported by a measurable symmetric subset $E\subset V\times V$. This applies to such diverse areas as optimization, graphons (limits of finite graphs), symbolic dynamics, measurable equivalence relations, to determinantal processes, to jump-processes; and it extends earlier studies of infinite graphs $G = (V, E)$ which are endowed with a symmetric weight function $c_{xy}$ defined on the set of edges $E$. As in the theory of weighted networks, we consider the Hilbert spaces $L^2(μ), L^2(cμ)$ and define two other Hilbert spaces, the dissipation space $Diss$ and finite energy space $\mathcal H_E$. Our main results include a number of explicit spectral theoretic and potential theoretic theorems that apply to two realizations of Laplace operators, and the associated jump-diffusion semigroups, one in $L^2(μ)$, and, the second, its counterpart in $\mathcal H_E$. We show in particular that it is the second setting (the energy-Hilbert space and the dissipation Hilbert space) which is needed in a detailed study of transient Markov processes.

math.FA↗

arXiv2017

Infinite-dimensional transfer operators, endomorphisms, and measurable partitions

We develop a new duality between endomorphisms of measure spaces, on the one hand, and a certain family of positive operators, called transfer operators, acting in spaces of measurable functions on, on the other. A framework of standard Borel spaces is adopted; and this generality is wide enough to cover a host of applications. While the mathematical structures of positive operators, endomorphisms, transfer operators, measurable partitions, and Markov processes arise in a host of settings, both pure and applied, we propose here a unified study. This is the general setting of dynamics in Borel measure spaces. Hence the corresponding linear structures are infinite-dimensional. Nonetheless, we prove a number of analogues of the more familiar finite-dimensional settings, for example, the Perron-Frobenius theorem for positive matrices, and the corresponding Markov chains. Tools from the theory of operators in Hilbert space of special significance to us will be the use of a certain universal Hilbert space, as well as classes of operators in it, directly related to the central theme of duality for transfer operators. From ergodic theory, we address such questions as measurable cross sections, partitions, and Rohlin analysis of endomorphisms of measure spaces. While there are classical theorems dealing with analogous questions for automorphisms of measure spaces, a systematic study of endomorphisms is of more recent vintage;-- in its infancy. In order to make the exposition accessible to students and to researchers in neighboring areas, we have included a number of explicit examples and applications.

math.FA↗

arXiv2016

Perfect orderings on Bratteli diagrams II: general Bratteli diagrams

We continue our study of orderings on Bratteli diagrams started in previous work, joint with Jan Kwiatkowski, where Bratteli diagrams of finite rank were considered. We extend the notions of languages, permutations (called correspondences in this paper), skeletons and associated graphs to the case of general Bratteli diagrams, and show their relevance to the study of perfect orderings: those that support Vershik maps; in particular, perfect orderings with several extremal paths. A perfect ordering comes equipped with a skeleton and a correspondence, and conversely, given a skeleton and correspondence, we describe explicitly how to construct perfect orderings, by showing that paths in the associated directed graphs determine the language of the order. We describe an explicit algorithmic method to create perfect orderings on Bratteli diagrams based on the study of certain relations between the entries of the diagram's incidence matrices and properties of the associated graphs, with the latter relations characterizing diagrams which support perfect orderings. Also, we apply the notions of skeletons and associated graphs, to give a new combinatorial proof of the fact that diagrams supporting perfect orderings with k maximal paths have a direct sum of k-1 copies of the integers contained in their infinitesimal subgroup. Under certain conditions, we show that a similar result holds if the diagram supports countably many maximal paths. Our results are illustrated by numerous examples.

math.DS↗