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- 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 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… (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)

In the paper we completely describe characters (central positive-definite functions) of simple locally finite groups that can be represented as inductive limits of (products of) symmetric groups under block diagonal embeddings. 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)

We prove that the only finite factor-representations of the HigmanThompson groups {Fn,r}, {Gn,r} are the regular representations and scalar representations arising from group abelianizations. As a corollary, we obtain that any measure-preserving ergodic action of a simple Higman-Thompson group must be essentially free. Finite factor representations of other… (More)

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

For a Cantor set X, let Homeo(X) denote the group of all homeomorphisms of X. The main result of this note is the following theorem. Let T ∈ Homeo(X) be an aperiodic homeomorphism, let μ1, μ2, . . . , μk be Borel probability measures on X, ε > 0, and n ≥ 2. Then there exists a clopen set E ⊂ X such that the sets E,TE, . . . , T n−1E are disjoint and… (More)

- COLLOQU IUM, MATHEMAT I CUM, K. MEDYNETS
- 2007

We consider the full group [φ] and topological full group [[φ]] of a Cantor minimal system (X, φ). We prove that the commutator subgroups D([φ]) and D([[φ]]) are simple and show that the groups D([φ]) and D([[φ]]) completely determine the class of orbit equivalence and flip conjugacy of φ, respectively. These results improve the classification found in… (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)

- K. Medynets
- 2010

In this note we consider dynamical systems (X, G) on a Cantor set X satisfying some mild technical conditions. The considered class includes, in particular, minimal and transitive aperiodic systems. We prove that two such systems (X1, G1) and (X2, G2) are orbit equivalent if and only if their full groups are isomorphic as abstract groups. This result is a… (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)

- Maŕıa Isabel Cortez, Konstantin Medynets
- 2015

The paper is focused on the study of continuous orbit equivalence for generalized odometers (profinite actions). We show that two generalized odometers are continuously orbit equivalent if and only if the acting groups have finite index subgroups (having the same index) whose actions are piecewise conjugate. This result extends M. Boyle’s flip-conjugacy… (More)