{𝑚_{𝑛}}-odometer and the binary odometer are finitarily orbit equivalent

2007 ◽  
pp. 123-134 ◽  
Author(s):  
Mrinal Kanti Roychowdhury
Keyword(s):  
Orbit Equivalent

scholarly journals L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1169-1188 ◽  
Author(s):  
ROMAN SAUER
Keyword(s):  
Probability Space ◽  
Betti Numbers ◽  
Free Action ◽  
Countable Group ◽  
Equivalence Relations ◽  
Dimension Theory ◽  
Discrete Groups ◽  
Type Ii ◽  
Orbit Equivalent ◽  
Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

scholarly journals The absorption theorem for affable equivalence relations

2008 ◽  
Vol 28 (5) ◽  
pp. 1509-1531 ◽  
Author(s):  
THIERRY GIORDANO ◽  
HIROKI MATUI ◽  
IAN F. PUTNAM ◽  
CHRISTIAN F. SKAU
Keyword(s):  
Dynamical Systems ◽  
Equivalence Relation ◽  
Closed Subset ◽  
Cantor Set ◽  
Equivalence Relations ◽  
Orbit Structure ◽  
Orbit Equivalent ◽  
Finite Set ◽  
Significant Generalization

AbstractWe prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being ‘small’ in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case—when Y is a finite set—this result is highly non-trivial. The result itself—called the absorption theorem—is a powerful and crucial tool for the study of the orbit structure of minimal ℤn-actions on the Cantor set, see Remark 4.8. The absorption theorem is a significant generalization of the main theorem proved in Giordano et al [Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys.24 (2004), 441–475] . However, we shall need a few key results from the above paper in order to prove the absorption theorem.

Topological equivalence and rigidity of flows on certain solvmanifolds

1999 ◽  
Vol 19 (3) ◽  
pp. 559-569
Author(s):  
D. BENARDETE ◽  
S. G. DANI
Keyword(s):  
Lie Group ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Real Vector ◽  
Orbit Equivalent ◽  
Finite Dimensional ◽  
Generic Class ◽  
Induced Flow ◽  
Simply Connected ◽  
Solvable Lie Group

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g) $, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers.Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.

Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows

2000 ◽  
Vol 20 (6) ◽  
pp. 1687-1710 ◽  
Author(s):  
RICHARD GJERDE ◽  
ØRJAN JOHANSEN
Keyword(s):  
Dimension Group ◽  
Orbit Equivalent ◽  
Bratteli Diagrams ◽  
Novel Approach ◽  
Dimension Groups ◽  
Substitution System ◽  
Zero Entropy ◽  
Factor Map ◽  
Cantor Minimal Systems ◽  
Uniquely Ergodic

We construct Bratteli–Vershik models for Toeplitz flows and characterize a class of properly ordered Bratteli diagrams corresponding to these flows. We use this result to extend by a novel approach—using basic theory of dimension groups—an interesting and non-trivial result about Toeplitz flows, first shown by Downarowicz. (Williams had previously obtained preliminary results in this direction.) The result states that to any Choquet simplex $K$, there exists a $0$–$1$ Toeplitz flow $(Y,\psi)$, so that the set of invariant probability measures of $(Y,\psi)$ is affinely homeomorphic to $K$. Not only do we give a conceptually new proof of this result, we also show that we may choose $(Y,\psi)$ to have zero entropy and to have full rational spectrum.Furthermore, our Bratteli–Vershik model for a given Toeplitz flow explicitly exhibits the factor map onto the maximal equicontinuous (odometer) factor. We utilize this to give a simple proof of the existence of a uniquely ergodic 0–1 Toeplitz flow of zero entropy having a given odometer as its maximal equicontinuous factor and being strongly orbit equivalent to this factor. By the same token, we show the existence of 0–1 Toeplitz flows having the 2-odometer as their maximal equicontinuous factor, being strong orbit equivalent to the same, and assuming any entropy value in $[0,\ln 2)$.Finally, we show by an explicit example, using Bratteli diagrams, that Toeplitz flows are not preserved under Kakutani equivalence (in fact, under inducing)—contrasting what is the case for substitution minimal systems. In fact, the example we exhibit is an induced system of a 0–1 Toeplitz flow which is conjugate to the Chacon substitution system, thus it is prime, i.e. it has no non-trivial factors.The thrust of our paper is to demonstrate the relevance and usefulness of Bratteli–Vershik models and dimension group theory for the study of minimal symbolic systems. This is also exemplified in recent papers by Forrest and by Durand, Host and Skau, treating substitution minimal systems, and by papers by Boyle, Handelman and by Ormes.

scholarly journals Lebesgue orbit equivalence of multidimensional Borel flows: A picturebook of tilings

2016 ◽  
Vol 37 (6) ◽  
pp. 1966-1996
Author(s):  
KONSTANTIN SLUTSKY
Keyword(s):  
Lebesgue Measure ◽  
Probability Measures ◽  
Orbit Equivalence ◽  
Orbit Equivalent

The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue orbit equivalence, by which we mean an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non-smooth $\mathbb{R}^{d}$-flows are shown to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.

scholarly journals Graph algebras and orbit equivalence

2015 ◽  
Vol 37 (2) ◽  
pp. 389-417 ◽  
Author(s):  
NATHAN BROWNLOWE ◽  
TOKE MEIER CARLSEN ◽  
MICHAEL F. WHITTAKER
Keyword(s):  
Directed Graphs ◽  
Orbit Equivalence ◽  
Graph Algebras ◽  
Orbit Equivalent ◽  
Markov Shifts ◽  
Graph Algebra ◽  
Arbitrary Graphs ◽  
Reverse Implication

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto’s notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their $C^{\ast }$-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs $E$ we construct a groupoid ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ from the graph algebra $C^{\ast }(E)$ and its diagonal subalgebra ${\mathcal{D}}(E)$ which generalises Renault’s Weyl groupoid construction applied to $(C^{\ast }(E),{\mathcal{D}}(E))$. We show that ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ recovers the graph groupoid ${\mathcal{G}}_{E}$ without the assumption that every cycle in $E$ has an exit, which is required to apply Renault’s results to $(C^{\ast }(E),{\mathcal{D}}(E))$. We finish with applications of our results to out-splittings of graphs and to amplified graphs.

scholarly journals Sofic entropy and amenable groups

2011 ◽  
Vol 32 (2) ◽  
pp. 427-466 ◽  
Author(s):  
LEWIS BOWEN
Keyword(s):  
Group Action ◽  
Amenable Group ◽  
Group Actions ◽  
Amenable Groups ◽  
Orbit Equivalent ◽  
Sofic Entropy ◽  
Classical Entropy

AbstractIn previous work, the author introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here, it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new measure-conjugacy invariant called upper-sofic entropy and a theorem of Rudolph and Weiss for the entropy of orbit-equivalent actions relative to the orbit changeσ-algebra.

scholarly journals Dynamical simplices and FraĂŻssĂŠ theory

2018 ◽  
Vol 39 (11) ◽  
pp. 3111-3126 ◽  
Author(s):  
JULIEN MELLERAY
Keyword(s):  
Invariant Measures ◽  
Probability Measures ◽  
Choquet Simplex ◽  
Cantor Space ◽  
Orbit Equivalent ◽  
Minimal Homeomorphisms

We simplify a criterion (due to Ibarlucía and the author) which characterizes dynamical simplices, that is, sets $K$ of probability measures on a Cantor space $X$ for which there exists a minimal homeomorphism of $X$ whose set of invariant measures coincides with $K$ . We then point out that this criterion is related to Fraïssé theory, and use that connection to provide a new proof of Downarowicz’ theorem stating that any non-empty metrizable Choquet simplex is affinely homeomorphic to a dynamical simplex. The construction enables us to prove that there exist minimal homeomorphisms of a Cantor space which are speedup equivalent but not orbit equivalent, answering a question of Ash.

scholarly journals Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions

2015 ◽  
Vol 36 (5) ◽  
pp. 1557-1581 ◽  
Author(s):  
KENGO MATSUMOTO ◽  
HIROKI MATUI
Keyword(s):  
Zeta Functions ◽  
Periodic Points ◽  
Orbit Equivalence ◽  
Cohomology Groups ◽  
Borel Measures ◽  
Orbit Equivalent ◽  
Positive Elements ◽  
Markov Shifts ◽  
Sigma B

For continuously orbit equivalent one-sided topological Markov shifts $(X_{A},{\it\sigma}_{A})$ and $(X_{B},{\it\sigma}_{B})$, their eventually periodic points and cocycle functions are studied. As a result, we directly construct an isomorphism between their ordered cohomology groups $(\bar{H}^{A},\bar{H}_{+}^{A})$ and $(\bar{H}^{B},\bar{H}_{+}^{B})$. We also show that the cocycle functions for the continuous orbit equivalences give rise to positive elements of their ordered cohomology groups, so that the zeta functions of continuously orbit equivalent topological Markov shifts are related. The set of Borel measures is shown to be invariant under continuous orbit equivalence of one-sided topological Markov shifts.

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