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S. Bezuglyi

Publications and source records attributed to S. Bezuglyi.

10 recordsLinked to original sources
arXiv2019

Exact number of ergodic invariant measures for Bratteli diagrams

For a Bratteli diagram $B$, we study the simplex $\mathcal{M}_1(B)$ of probability measures on the path space of $B$ which are invariant with respect to the tail equivalence relation. Equivalently, $\mathcal{M}_1(B)$ 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 $\mathcal{M}_1(B)$ 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 $\mathcal{M}_1(B)$ is a singleton. For a finite rank $k$ Bratteli diagram $B$ having exactly $l \leq k$ 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.

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arXiv2019

Invariant measures for Cantor dynamical systems

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.

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arXiv2015

Bratteli diagrams: structure, measures, dynamics

This paper is a survey on general (simple and non-simple) Bratteli diagrams which focuses on the following topics: finite and infinite tail invariant measures on the path space $X_B$ of a Bratteli diagram $B$, existence of continuous dynamics on $X_B$ compatible with tail equivalence relation, subdiagrams and measure supports. We also discuss the structure of Bratteli diagrams, orbit equivalence and full groups, homeomorphic measures.

math.DS↗
arXiv2015

Subdiagrams and invariant measures on Bratteli diagrams

We study ergodic finite and infinite measures defined on the path space $X_B$ of a Bratteli diagram $B$ which are invariant with respect to the tail equivalence relation on $X_B$. Our interest is focused on measures supported by vertex and edge subdiagrams of $B$. We give several criteria when a finite invariant measure defined on the path space of a subdiagram of $B$ extends to a finite invariant measure on $B$. Given a finite ergodic measure on a Bratteli diagram $B$ and a subdiagram $B'$ of $B$, we find the necessary and sufficient conditions under which the measure of the path space $X_{B'}$ of $B'$ is positive. For a class of Bratteli diagrams of finite rank, we determine when they have maximal possible number of ergodic invariant measures. The case of diagrams of rank two is completely studied. We include also an example which explicitly illustrates the proved results.

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arXiv2012

Orbit equivalent substitution dynamical systems and complexity

For any primitive proper substitution σ, we give explicit constructions of countably many pairwise non-isomorphic substitution dynamical systems {(X_{ζ_n}, T_{ζ_n})}_{n=1}^{\infty} such that they all are (strong) orbit equivalent to (X_σ, T_σ). We show that the complexity of the substitution dynamical systems {(X_{ζ_n}, T_{ζ_n})} 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).

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arXiv2010

Homeomorphic measures on stationary Bratteli diagrams

We study the set S of ergodic probability Borel measures on stationary non-simple Bratteli diagrams which are invariant with respect to the tail equivalence relation. Equivalently, the set S is formed by ergodic probability measures invariant with respect to aperiodic substitution dynamical systems. The paper is devoted to the classification of measures $μ$ from S with respect to a homeomorphism. The properties of these measures related to the clopen values set $S(μ)$ are studied. It is shown that for every measure $μ$ in S there exists a subgroup G of $\mathbb R$ such that $S(μ)$ is the intersection of G with [0,1], i.e. $S(μ)$ is group-like. A criterion of goodness is proved for such measures. This result is used to classify the measures from S up to a homeomorphism. It is proved that for every good measure $μ$ in S there exist countably many measures $\{μ_i\}_{i\in \mathbb N}$ from S such that $μ$ and $μ_i$ are homeomorphic measures but the tail equivalence relations on corresponding Bratteli diagrams are not orbit equivalent.

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arXiv2009

Invariant Measures on Stationary Bratteli Diagrams

We study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we explicitly describe all ergodic probability measures invariant with respect to the tail equivalence relation (or the Vershik map). These measures are completely described by the incidence matrix of the diagram. Since such diagrams correspond to substitution dynamical systems, this description gives an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.

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arXiv2007

Aperiodic substitutional systems and their Bratteli diagrams

In the paper we study aperiodic substitutional dynamical systems arisen from non-primitive substitutions. We prove that the Vershik homeomorphism $ϕ$ of a stationary ordered Bratteli diagram is homeomorphic to an aperiodic substitutional system if and only if no restriction of $ϕ$ to a minimal component is homeomorphic to an odometer. We also show that every aperiodic substitutional system generated by a substitution with nesting property is homeomorphic to the Vershik map of a stationary ordered Bratteli diagram. It is proved that every aperiodic substitutional system is recognizable. The classes of $m$-primitive substitutions and associated to them derivative substitutions are studied. We discuss also the notion of expansiveness for Cantor dynamical systems of finite rank.

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arXiv2005

Approximation in ergodic theory, Borel, and Cantor dynamics

This survey is focused on the results related to topologies on the groups of transformations in ergodic theory, Borel, and Cantor dynamics. Various topological properties (density, connectedness, genericity) of these groups and their subsets (subgroups) are studied.

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