Related papers: Full Groups and Orbit Equivalence in Cantor Dynami…

Previous work showed that every pair of nontrivial Bernoulli shifts over a fixed free group are orbit equivalent. In this paper, we prove that if $G_1,G_2$ are nonabelian free groups of finite rank then every nontrivial Bernoulli shift over…

We study the relationship between pairs of topological dynamical systems $ (X,T) $ and $ (X',T') $, where $ (X',T') $ is the quotient of $ (X,T) $ under the action of a finite group $ G $. We describe three phenomena concerning the…

We introduce notions of continuous orbit equivalence and strong (respective, weak) continuous orbit equivalence for automorphism systems of \'{e}tale equivalence relations, and characterize them in terms of the semi-direct product…

In the paper we discuss the algebraic structure of topological full group $[[T]]$ of a Cantor minimal system $(X,T)$. We show that the topological full group $[[T]]$ has the structure similar to a union of permutational wreath products of…

Cohomology for actions of free abelian groups on the Cantor set has (when endowed with an order structure) provided a complete invariance for orbit equivalence. In this paper, we study a particular class of actions of such groups called…

Let $G_\Gamma\curvearrowright X$ and $G_\Lambda\curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every…

We construct a minimal topological action $\wp$ of a non-amenable group on a Cantor set $C$, which is non-uniquely ergodic and furthermore there exist ergodic invariant measures $\mu_1$ and $\mu_2$ such that $(\wp,C,\mu_1)$ and…

By a Cantor group we mean a topological group homeomorphic to the Cantor set. The author earlier proved that every compact metric space of rational cohomological dimension n can be obtained as the orbit space of a Cantor group action on a…

Let $A$ be a unital simple C*-algebra with tracial rank zero and $X$ be a compact metric space. Suppose that $h_1, h_2: C(X)\to A$ are two unital monomorphisms. We show that $h_1$ and $h_2$ are approximately unitarily equivalent if and only…

We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $\Gamma$ on a topological manifold where the…

We prove that commutator subgroups of topological full groups arising from minimal subshifts have exponential growth. We also prove that the measurable full group associated to the countable, measure-preserving, ergodic and hyperfinite…

We show that in the neighborhood of each ``finite type'' singular orbit of a real analytic integrable dynamical system (hamiltonian or not) there is a real analytic torus action which preserves the system and which is transitive on this…

We prove that a connected locally compact median space of finite rank which admits a transitive action is isometric to $\mathbb{R}^n$ endowed with the $\ell^1$-metric. In the other side, replacing the transitivity assumption on the group of…

A $\lambda$-graph system ${\frak L}$ is a generalization of a finite labeled graph and presents a subshift. We will prove that the topological dynamical systems $(X_{{\frak L}_1},\sigma_{{\frak L}_1})$ and $(X_{{\frak L}_2},\sigma_{{\frak…

Several versions of approximate conjugacy for minimal dynamical systems are introduced. Relation between approximate conjugacy and corresponding crossed product $C^*$-algebras is discussed. For the Cantor minimal systems, a complete…

We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system…

We show that for a countable discrete group which is locally of finite asymptotic dimension, the generic continuous action on Cantor space has hyperfinite orbit equivalence relation. In particular, this holds for free groups, answering a…

An example of an infinite regular feebly compact quasitopological group is presented such that all continuous real-valued functions on the group are constant. The example is based on the use of Korovin orbits in $X^G$, where $X$ is a…

Quasi-invariant measures for non-discrete groups of diffeomorphisms containing a Morse-Smale dynamics are studied. The assumption concerning the presence of a Morse-Smale dynamics allows us to extend to higher dimensions a number of…

We prove that any ergodic measure-preserving action of an irreducible lattice in a semisimple group, with finite center and each simple factor having rank at least two, either has finite orbits or has finite stabilizers. The same dichotomy…