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Publications (23)
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 a...
We introduce stationary generalized Bratteli diagrams $B$ which are represented as the union of countably many classical Pascal-Bratteli diagrams. We describe all ergodic invariant measures on $B$. We show that there exist orders which produce no infinite minimal or maximal paths and the corresponding Vershik map is a homeomorphism. We also describ...
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 provid...
Bratteli-Vershik models have been very successfully applied to the study of dynamics on compact metric spaces and, 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 man...
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 $(X,T)$...
We study the simplex $\mathcal{M}_1(B)$ of probability measures on a Bratteli diagram $B$ 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 $k$ Bratteli diagram $B$ has $l \leq k$ ergodic invariant measures, we describe the structures of the...
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...
In this paper we focus on Bratteli-Vershik models of general compact zero-dimensional systems with the action of a homeomorphism. An ordered Bratteli diagram is called decisive if the corresponding Vershik map prolongs in a unique way to a homeomorphism of the whole path space of the Bratteli diagram. We prove that a compact invertible zero-dimensi...
In this paper we focus on Bratteli-Vershik models of general compact zero-dimensional systems with the action of a homeomorphism. An ordered Bratteli diagram is called decisive if the corresponding Vershik map prolongs in a unique way to a homeomorphism of the whole path space of the Bratteli diagram. We prove that a compact invertible zero-dimensi...
This paper is based on lecture notes of a course delivered in spring 2016 for PhD students. The goal of the course was to put together several techniques in handling dynamical systems on zero-dimensional spaces, such as uniform generators, symbolic extensions with an embedding, inverse limit representation, array representation, Bratteli-Vershik re...
The goal of this paper is to put together several techniques in handling dynamical systems on zero-dimensional spaces, such as array representation, inverse limit representation, or Bratteli-Vershik representation. We describe how one can switch from one representation to another. We also briefly review some more recent related notions: symbolic ex...
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 s...
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...
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...
We study finite measures on Bratteli diagrams invariant with respect to the
tail equivalence relation. Amongst the proved results on finiteness of measure
extension, we characterize the vertices of a Bratteli diagram that support an
ergodic finite invariant measure.
We study the set M(X) of full non-atomic Borel (finite or infinite) measures
on a non-compact locally compact Cantor set X. For an infinite measure $\mu$ in
M(X), the set $\mathfrak{M}_\mu = \{x \in X : {for any compact open set} U \ni
x {we have} \mu(U) = \infty \}$ is called defective. We call $\mu$
non-defective if $\mu(\mathfrak{M}_\mu) = 0$. T...
For any primitive proper substitution \sigma, we give explicit constructions
of countably many pairwise non-isomorphic substitution dynamical systems
{(X_{\zeta_n}, T_{\zeta_n})}_{n=1}^{\infty} such that they all are (strong)
orbit equivalent to (X_{\sigma}, T_{\sigma}). We show that the complexity of
the substitution dynamical systems {(X_{\zeta_n...
We study the set $M_\infty(X)$ of all infinite full non-atomic Borel measures
on a Cantor space X. For a measure $\mu$ from $M_\infty(X)$ we define a
defective set $M_\mu = \{x \in X : for any clopen set U which contains x we
have \mu(U) = \infty \}$. We call a measure $\mu$ from $M_\infty(X)$
non-defective ($\mu \in M_\infty^0(X)$) if $\mu(M_\mu)...
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 mea...