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In the paper we completely describe characters (central positive-definite functions) of simple locally finite groups that can be represented as induc-tive limits of (products of) symmetric groups under block diagonal embed-dings. Each such group G defines an infinite graded graph that encodes the embedding scheme. The group G acts on the space X of infinite… (More)

- S Bezuglyi, A H Dooley, K Medynets
- 2005

For a Cantor set X, let Homeo(X) denote the group of all homeo-morphisms of X. The main result of this note is the following theorem. Several corollaries of this result are given. In particular, it is proved that for any aperiodic T ∈ Homeo(X) the set of all homeomorphisms conjugate to T is dense in the set of aperiodic homeomorphisms. 0. Introduction. One… (More)

- S Bezuglyi, J Kwiatkowski, K Medynets
- 2008

In the paper we study aperiodic substitutional dynamical systems arisen from non-primitive substitutions. We prove that the Vershik home-omorphism 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… (More)

- S Bezuglyi, K Medynets
- 2004

Let Aut(X, B) be the group of all Borel automorphisms of a standard Borel space (X, B). We study topological properties of Aut(X, B) with respect to the uniform and weak topologies, τ and p, defined in [BDK1]. It is proved that the class of smooth automorphisms is dense in (Aut(X, B), p). Let Ctbl(X) denote the group of Borel automorphisms with countable… (More)

We continue to study topological properties of the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the uniform topology τ , which was started in [B-K 1], [B-D-K 1; 2], [B-D-M], and [M]. We prove that the set of periodic homeomorphisms is τ-dense in Homeo(X) and deduce from this result that the topological group (Homeo(X), τ) has the… (More)

- S. Bezuglyi, J. Kwiatkowski, K. Medynets, K. MEDYNETS
- 2004

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. In this paper, we intend to present a unified approach to the study of topological… (More)

- S. Bezuglyi, J. Kwiatkowski, K. Medynets, B. Solomyak
- 2010

In this paper we study ergodic measures on non-simple Bratteli diagrams of finite rank that are invariant with respect to the cofinal equivalence relation. We describe the structure of finite rank diagrams and prove that every ergodic invariant measure (finite or infinite) is an extension of a finite ergodic measure defined on a simple subdiagram. We find… (More)

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