Sergey Bezuglyi | The University of Iowa (original) (raw)

Papers by Sergey Bezuglyi

Research paper thumbnail of New Hilbert Space Tools for Analysis of Graph Laplacians and Markov Processes

Complex Analysis and Operator Theory, Sep 20, 2023

Research paper thumbnail of Finite Rank Bratteli Diagrams: Structure of Invariant Measures

arXiv (Cornell University), Mar 14, 2010

We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measu... more We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or "regular" infinite) is obtained by an extension from a simple subdiagram. We further investigate quantitative properties of these measures, which are mainly determined by the asymptotic behavior of products of incidence matrices. A number of sufficient conditions for unique ergodicity are obtained. One of these is a condition of exact finite rank, which parallels a similar notion in measurable dynamics. Several examples illustrate the broad range of possible behavior of finite type diagrams and invariant measures on them. We then prove that the Vershik map on the path space of an exact finite rank diagram cannot be strongly mixing, independent of the ordering. On the other hand, for the so-called "consecutive" ordering, the Vershik map is not strongly mixing on all finite rank diagrams.

Research paper thumbnail of Finite Rank Bratteli Diagrams: Structure of Invariant Measures

arXiv (Cornell University), Mar 14, 2010

We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measu... more We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or "regular" infinite) is obtained by an extension from a simple subdiagram. We further investigate quantitative properties of these measures, which are mainly determined by the asymptotic behavior of products of incidence matrices. A number of sufficient conditions for unique ergodicity are obtained. One of these is a condition of exact finite rank, which parallels a similar notion in measurable dynamics. Several examples illustrate the broad range of possible behavior of finite type diagrams and invariant measures on them. We then prove that the Vershik map on the path space of an exact finite rank diagram cannot be strongly mixing, independent of the ordering. On the other hand, for the so-called "consecutive" ordering, the Vershik map is not strongly mixing on all finite rank diagrams.

Research paper thumbnail of Graph Laplace and Markov operators on a measure space

arXiv (Cornell University), Jan 13, 2018

The main goal of this paper is to build a measurable analogue to the theory of weighted networks ... more 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, B, µ) and a symmetric measure ρ on (V × V, B × B) supported by a measurable symmetric subset E ⊂ V × V. This applies to such diverse areas as optimization, graphons (limits of finite graphs), symbolic dynamics, measurable equivalence relations, to determinantal processes, to jumpprocesses; and it extends earlier studies of infinite graphs G = (V, E) which are endowed with a symmetric weight function cxy 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 HE. 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 HE. 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.

Research paper thumbnail of Measures on Cantor sets: the good, the ugly, the bad

arXiv (Cornell University), Jan 9, 2012

We translate Akin's notion of good (and related concepts) from measures on Cantor sets to traces ... more We translate Akin's notion of good (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups), this yields characterizations and examples, which translate back to the original context. Good traces on a simple dimension group are characterized by their kernel having dense image in their annihilating set of affine functions on the trace space; this makes it possible to construct many examples with seemingly paradoxical properties. In order to study the related property of refinability, we consider goodness for sets of measures (traces on dimension groups), and obtain partial characterizations in terms of (special) convex subsets of Choquet simplices. These notions also very closely related to unperforation of quotients of dimension groups by convex subgroups (that are not order ideals), and we give partial characterizations. Numerous examples illustrate the results. Sergey Bezuglyi & David Handelman 1 * This is the correct translation of the title of the Italian spaghetti western (Il buono, il brutto, il cattivo, D: Sergio Leone), which was changed for US audiences. The English language title, The good, the bad, and the ugly, is clichéd now-over 175 articles in engineering and mathematics use it in their title-but we could not find any titles using the original order. 1 Supported in part by an NSERC Discovery Grant.

Research paper thumbnail of Invariant measures for Cantor dynamical systems

arXiv (Cornell University), Apr 21, 2019

This paper is a survey devoted to the study of probability and infinite ergodic invariant measure... more This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure or finitely many such measures (finitely ergodic homeomorphisms). Since every Cantor dynamical system (X, T) can be realized as a Vershik map acting on the path space of a Bratteli diagram, we use combinatorial methods developed in symbolic dynamics and Bratteli diagrams during the last decade to study the simplex of invariant measures.

Research paper thumbnail of Exact number of ergodic invariant measures for Bratteli diagrams

arXiv (Cornell University), Aug 31, 2017

For a Bratteli diagram B, we study the simplex M1(B) of probability measures on the path space of... more For a Bratteli diagram B, we study the simplex M1(B) of probability measures on the path space of B which are invariant with respect to the tail equivalence relation. Equivalently, M1(B) is formed by probability measures invariant with respect to a homeomorphism of a Cantor set. We study relations between the number of ergodic measures from M1(B) and the structure and properties of the diagram B. We prove a criterion and find sufficient conditions of unique ergodicity of a Bratteli diagram, in which case the simplex M1(B) is a singleton. For a finite rank k Bratteli diagram B having exactly l ≤ k ergodic invariant measures, we explicitly describe the structure of the diagram and find the subdiagrams which support these measures. We find sufficient conditions under which: (i) a Bratteli diagram has a prescribed number (finite or infinite) of ergodic invariant measures, and (ii) the extension of a measure from a uniquely ergodic subdiagram gives a finite ergodic invariant measure. Several examples, including stationary Bratteli diagrams, Pascal-Bratteli diagrams, and Toeplitz flows, are considered. Contents

Research paper thumbnail of Measurable multiresolution systems, endomorphisms, and representations of Cuntz relations

arXiv (Cornell University), Apr 27, 2023

The purpose of this paper is to present new classes of function systems as part of multiresolutio... more 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.

Research paper thumbnail of Orbit equivalent substitution dynamical systems and complexity

arXiv (Cornell University), Jan 8, 2012

For any primitive proper substitution σ, we give explicit constructions of countably many pairwis... more For any primitive proper substitution σ, we give explicit constructions of countably many pairwise non-isomorphic substitution dynamical systems {(X ζn , T ζn)} ∞ n=1 such that they all are (strong) orbit equivalent to (X σ , T σ). We show that the complexity of the substitution dynamical systems {(X ζn , T ζn)} is essentially different that prevents them from being isomorphic. Given a primitive (not necessarily proper) substitution τ , we find a stationary simple properly ordered Bratteli diagram with the least possible number of vertices such that the corresponding Bratteli-Vershik system is orbit equivalent to (X τ , T τ).

Research paper thumbnail of Exact number of ergodic measures for Bratteli diagrams

arXiv (Cornell University), Aug 31, 2017

We study the simplex mathcalM_1(B)\mathcal{M}_1(B)mathcalM1(B) of probability measures on a Bratteli diagram BBB which a... more We study the simplex mathcalM1(B)\mathcal{M}_1(B)mathcalM_1(B) of probability measures on a Bratteli diagram BBB which are invariant with respect to the tail equivalence relation. We prove a criterion of unique ergodicity of a Bratteli diagram. In case when a finite rank kkk Bratteli diagram BBB has lleqkl \leq klleqk ergodic invariant measures, we describe the structures of the diagram and the subdiagrams which support these measures. We find conditions under which the extension of a measure from a uniquely ergodic subdiagram is a finite ergodic measure.

Research paper thumbnail of Alexeyev's photo

Cambridge University Press eBooks, Dec 8, 2003

Research paper thumbnail of Monopoles, dipoles, and harmonic functions on Bratteli diagrams

arXiv (Cornell University), Aug 5, 2015

In our study of electrical networks we develop two themes: finding explicit formulas for special ... more In our study of electrical networks we develop two themes: finding explicit formulas for special classes of functions defined on the vertices of a transient network, namely monopoles, dipoles, and harmonic functions. Secondly, our interest is focused on the properties of electrical networks supported on Bratteli diagrams. We show that the structure of Bratteli diagrams allows one to describe algorithmically harmonic functions as well as monopoles and dipoles. We also discuss some special classes of Bratteli diagrams (stationary, Pascal, trees), and we give conditions under which the harmonic functions defined on these diagrams have finite energy.

Research paper thumbnail of Examples

Lecture Notes in Mathematics, 2018

Research paper thumbnail of Endomorphisms and Measurable Partitions

Lecture Notes in Mathematics, 2018

In this chapter, we collect definitions and some basic facts about the underlying spaces, endomor... more In this chapter, we collect definitions and some basic facts about the underlying spaces, endomorphisms, measurable partitions, etc., which are used throughout the book. Though these notions are known in ergodic theory, we discuss them for the reader’s convenience.

Research paper thumbnail of On cocycles of ergodic dynamical systems and automorphisms compatible with them

Advances in Soviet mathematics, Oct 5, 1994

Research paper thumbnail of Bratteli diagrams in Borel dynamics

arXiv (Cornell University), Dec 28, 2022

Bratteli-Vershik models have been very successfully applied to the study of various dynamical sys... more 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 noncompact 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 tailinvariant 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.

Research paper thumbnail of IFS measures on generalized Bratteli diagrams

Cornell University - arXiv, Oct 25, 2022

The purpose of the paper is a general analysis of path space measures. Our focus is a certain pat... more 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.

Research paper thumbnail of Transfer Operators on the Space of Densities

Transfer Operators, Endomorphisms, and Measurable Partitions, 2018

This chapter is focused on the study of an important class of transfer operators. As usual, we fi... more This chapter is focused on the study of an important class of transfer operators. As usual, we fix a non-invertible non-singular dynamical system (X, B, μ, σ). Without loss of generality, we can assume that μ is a finite (even probability) measure because μ can be replaced by any measure equivalent to μ. Keywords Non-invertible non-singular dynamical systems • Quasi-invariant measures We recall that if λ is a Borel measure such that λ μ, then there exists the Radon-Nikodym derivative f (x) = dλ dμ (x). Conversely, any nonnegative function f ∈ L 1 (μ) serves as a density function for a measure dλ = f dμ. Definition 10.1 Define a transfer operator R μ = (R, σ) acting on L 1 (μ) by the formula R μ (f)(x) = (f dμ) • σ −1 dμ (x), f ∈ L 1 (μ). (10.1) We call R μ a transfer operator on the space of densities. In this chapter, we will work only with transfer operators R μ defined by (10.1). The following lemma contains main properties of R = R μ. Most of the statements are well known, so that we omit their proofs. Lemma 10.2 Let R be defined by (10.1) where the measure μ is quasi-invariant with respect to σ. The following statements hold.

Research paper thumbnail of Introduction and Examples

We present a unified study a class of positive operators called (generalized) transfer operators,... more We present a unified study a class of positive operators called (generalized) transfer operators, and of their applications to the study of endomorphisms, measurable partitions, and Markov processes, as they arise in diverse settings. We begin with the setting of dynamics in standard Borel, and measure, spaces.

Research paper thumbnail of Wold’s Theorem and Automorphic Factors of Endomorphisms

In this chapter, we discuss Wold’s theorem stating the existence of a decomposition of any isomet... more In this chapter, we discuss Wold’s theorem stating the existence of a decomposition of any isometry operator of a Hilbert space in a unitary part and a unilateral shift.

Research paper thumbnail of New Hilbert Space Tools for Analysis of Graph Laplacians and Markov Processes

Complex Analysis and Operator Theory, Sep 20, 2023

Research paper thumbnail of Finite Rank Bratteli Diagrams: Structure of Invariant Measures

arXiv (Cornell University), Mar 14, 2010

We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measu... more We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or "regular" infinite) is obtained by an extension from a simple subdiagram. We further investigate quantitative properties of these measures, which are mainly determined by the asymptotic behavior of products of incidence matrices. A number of sufficient conditions for unique ergodicity are obtained. One of these is a condition of exact finite rank, which parallels a similar notion in measurable dynamics. Several examples illustrate the broad range of possible behavior of finite type diagrams and invariant measures on them. We then prove that the Vershik map on the path space of an exact finite rank diagram cannot be strongly mixing, independent of the ordering. On the other hand, for the so-called "consecutive" ordering, the Vershik map is not strongly mixing on all finite rank diagrams.

Research paper thumbnail of Finite Rank Bratteli Diagrams: Structure of Invariant Measures

arXiv (Cornell University), Mar 14, 2010

We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measu... more We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or "regular" infinite) is obtained by an extension from a simple subdiagram. We further investigate quantitative properties of these measures, which are mainly determined by the asymptotic behavior of products of incidence matrices. A number of sufficient conditions for unique ergodicity are obtained. One of these is a condition of exact finite rank, which parallels a similar notion in measurable dynamics. Several examples illustrate the broad range of possible behavior of finite type diagrams and invariant measures on them. We then prove that the Vershik map on the path space of an exact finite rank diagram cannot be strongly mixing, independent of the ordering. On the other hand, for the so-called "consecutive" ordering, the Vershik map is not strongly mixing on all finite rank diagrams.

Research paper thumbnail of Graph Laplace and Markov operators on a measure space

arXiv (Cornell University), Jan 13, 2018

The main goal of this paper is to build a measurable analogue to the theory of weighted networks ... more 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, B, µ) and a symmetric measure ρ on (V × V, B × B) supported by a measurable symmetric subset E ⊂ V × V. This applies to such diverse areas as optimization, graphons (limits of finite graphs), symbolic dynamics, measurable equivalence relations, to determinantal processes, to jumpprocesses; and it extends earlier studies of infinite graphs G = (V, E) which are endowed with a symmetric weight function cxy 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 HE. 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 HE. 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.

Research paper thumbnail of Measures on Cantor sets: the good, the ugly, the bad

arXiv (Cornell University), Jan 9, 2012

We translate Akin's notion of good (and related concepts) from measures on Cantor sets to traces ... more We translate Akin's notion of good (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups), this yields characterizations and examples, which translate back to the original context. Good traces on a simple dimension group are characterized by their kernel having dense image in their annihilating set of affine functions on the trace space; this makes it possible to construct many examples with seemingly paradoxical properties. In order to study the related property of refinability, we consider goodness for sets of measures (traces on dimension groups), and obtain partial characterizations in terms of (special) convex subsets of Choquet simplices. These notions also very closely related to unperforation of quotients of dimension groups by convex subgroups (that are not order ideals), and we give partial characterizations. Numerous examples illustrate the results. Sergey Bezuglyi & David Handelman 1 * This is the correct translation of the title of the Italian spaghetti western (Il buono, il brutto, il cattivo, D: Sergio Leone), which was changed for US audiences. The English language title, The good, the bad, and the ugly, is clichéd now-over 175 articles in engineering and mathematics use it in their title-but we could not find any titles using the original order. 1 Supported in part by an NSERC Discovery Grant.

Research paper thumbnail of Invariant measures for Cantor dynamical systems

arXiv (Cornell University), Apr 21, 2019

This paper is a survey devoted to the study of probability and infinite ergodic invariant measure... more This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure or finitely many such measures (finitely ergodic homeomorphisms). Since every Cantor dynamical system (X, T) can be realized as a Vershik map acting on the path space of a Bratteli diagram, we use combinatorial methods developed in symbolic dynamics and Bratteli diagrams during the last decade to study the simplex of invariant measures.

Research paper thumbnail of Exact number of ergodic invariant measures for Bratteli diagrams

arXiv (Cornell University), Aug 31, 2017

For a Bratteli diagram B, we study the simplex M1(B) of probability measures on the path space of... more For a Bratteli diagram B, we study the simplex M1(B) of probability measures on the path space of B which are invariant with respect to the tail equivalence relation. Equivalently, M1(B) is formed by probability measures invariant with respect to a homeomorphism of a Cantor set. We study relations between the number of ergodic measures from M1(B) and the structure and properties of the diagram B. We prove a criterion and find sufficient conditions of unique ergodicity of a Bratteli diagram, in which case the simplex M1(B) is a singleton. For a finite rank k Bratteli diagram B having exactly l ≤ k ergodic invariant measures, we explicitly describe the structure of the diagram and find the subdiagrams which support these measures. We find sufficient conditions under which: (i) a Bratteli diagram has a prescribed number (finite or infinite) of ergodic invariant measures, and (ii) the extension of a measure from a uniquely ergodic subdiagram gives a finite ergodic invariant measure. Several examples, including stationary Bratteli diagrams, Pascal-Bratteli diagrams, and Toeplitz flows, are considered. Contents

Research paper thumbnail of Measurable multiresolution systems, endomorphisms, and representations of Cuntz relations

arXiv (Cornell University), Apr 27, 2023

The purpose of this paper is to present new classes of function systems as part of multiresolutio... more 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.

Research paper thumbnail of Orbit equivalent substitution dynamical systems and complexity

arXiv (Cornell University), Jan 8, 2012

For any primitive proper substitution σ, we give explicit constructions of countably many pairwis... more For any primitive proper substitution σ, we give explicit constructions of countably many pairwise non-isomorphic substitution dynamical systems {(X ζn , T ζn)} ∞ n=1 such that they all are (strong) orbit equivalent to (X σ , T σ). We show that the complexity of the substitution dynamical systems {(X ζn , T ζn)} is essentially different that prevents them from being isomorphic. Given a primitive (not necessarily proper) substitution τ , we find a stationary simple properly ordered Bratteli diagram with the least possible number of vertices such that the corresponding Bratteli-Vershik system is orbit equivalent to (X τ , T τ).

Research paper thumbnail of Exact number of ergodic measures for Bratteli diagrams

arXiv (Cornell University), Aug 31, 2017

We study the simplex mathcalM_1(B)\mathcal{M}_1(B)mathcalM1(B) of probability measures on a Bratteli diagram BBB which a... more We study the simplex mathcalM1(B)\mathcal{M}_1(B)mathcalM_1(B) of probability measures on a Bratteli diagram BBB which are invariant with respect to the tail equivalence relation. We prove a criterion of unique ergodicity of a Bratteli diagram. In case when a finite rank kkk Bratteli diagram BBB has lleqkl \leq klleqk ergodic invariant measures, we describe the structures of the diagram and the subdiagrams which support these measures. We find conditions under which the extension of a measure from a uniquely ergodic subdiagram is a finite ergodic measure.

Research paper thumbnail of Alexeyev's photo

Cambridge University Press eBooks, Dec 8, 2003

Research paper thumbnail of Monopoles, dipoles, and harmonic functions on Bratteli diagrams

arXiv (Cornell University), Aug 5, 2015

In our study of electrical networks we develop two themes: finding explicit formulas for special ... more In our study of electrical networks we develop two themes: finding explicit formulas for special classes of functions defined on the vertices of a transient network, namely monopoles, dipoles, and harmonic functions. Secondly, our interest is focused on the properties of electrical networks supported on Bratteli diagrams. We show that the structure of Bratteli diagrams allows one to describe algorithmically harmonic functions as well as monopoles and dipoles. We also discuss some special classes of Bratteli diagrams (stationary, Pascal, trees), and we give conditions under which the harmonic functions defined on these diagrams have finite energy.

Research paper thumbnail of Examples

Lecture Notes in Mathematics, 2018

Research paper thumbnail of Endomorphisms and Measurable Partitions

Lecture Notes in Mathematics, 2018

In this chapter, we collect definitions and some basic facts about the underlying spaces, endomor... more In this chapter, we collect definitions and some basic facts about the underlying spaces, endomorphisms, measurable partitions, etc., which are used throughout the book. Though these notions are known in ergodic theory, we discuss them for the reader’s convenience.

Research paper thumbnail of On cocycles of ergodic dynamical systems and automorphisms compatible with them

Advances in Soviet mathematics, Oct 5, 1994

Research paper thumbnail of Bratteli diagrams in Borel dynamics

arXiv (Cornell University), Dec 28, 2022

Bratteli-Vershik models have been very successfully applied to the study of various dynamical sys... more 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 noncompact 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 tailinvariant 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.

Research paper thumbnail of IFS measures on generalized Bratteli diagrams

Cornell University - arXiv, Oct 25, 2022

The purpose of the paper is a general analysis of path space measures. Our focus is a certain pat... more 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.

Research paper thumbnail of Transfer Operators on the Space of Densities

Transfer Operators, Endomorphisms, and Measurable Partitions, 2018

This chapter is focused on the study of an important class of transfer operators. As usual, we fi... more This chapter is focused on the study of an important class of transfer operators. As usual, we fix a non-invertible non-singular dynamical system (X, B, μ, σ). Without loss of generality, we can assume that μ is a finite (even probability) measure because μ can be replaced by any measure equivalent to μ. Keywords Non-invertible non-singular dynamical systems • Quasi-invariant measures We recall that if λ is a Borel measure such that λ μ, then there exists the Radon-Nikodym derivative f (x) = dλ dμ (x). Conversely, any nonnegative function f ∈ L 1 (μ) serves as a density function for a measure dλ = f dμ. Definition 10.1 Define a transfer operator R μ = (R, σ) acting on L 1 (μ) by the formula R μ (f)(x) = (f dμ) • σ −1 dμ (x), f ∈ L 1 (μ). (10.1) We call R μ a transfer operator on the space of densities. In this chapter, we will work only with transfer operators R μ defined by (10.1). The following lemma contains main properties of R = R μ. Most of the statements are well known, so that we omit their proofs. Lemma 10.2 Let R be defined by (10.1) where the measure μ is quasi-invariant with respect to σ. The following statements hold.

Research paper thumbnail of Introduction and Examples

We present a unified study a class of positive operators called (generalized) transfer operators,... more We present a unified study a class of positive operators called (generalized) transfer operators, and of their applications to the study of endomorphisms, measurable partitions, and Markov processes, as they arise in diverse settings. We begin with the setting of dynamics in standard Borel, and measure, spaces.

Research paper thumbnail of Wold’s Theorem and Automorphic Factors of Endomorphisms

In this chapter, we discuss Wold’s theorem stating the existence of a decomposition of any isomet... more In this chapter, we discuss Wold’s theorem stating the existence of a decomposition of any isometry operator of a Hilbert space in a unitary part and a unilateral shift.