scholarly journals Orbit equivalence of graphs and isomorphism of graph groupoids

2018 ◽  
Vol 123 (2) ◽  
pp. 239-248 ◽  
Author(s):  
Toke Meier Carlsen ◽  
Marius Lie Winger
Keyword(s):  
Directed Graphs ◽  
Periodic Points ◽  
Path Space ◽  
Orbit Equivalence ◽  
Graph Groupoids ◽  
Orbit Equivalent ◽  
Boundary Path ◽  
Isolated Points

We show that the groupoids of two directed graphs are isomorphic if and only if the two graphs are orbit equivalent by an orbit equivalence that preserves isolated eventually periodic points. We also give a complete description of the (topological) isolated points of the boundary path space of a graph. As a result, we are able to show that the groupoids of two directed graphs with finitely many vertices and no sinks are isomorphic if and only if the two graphs are orbit equivalent, and that the groupoids of the stabilisations of two such graphs are isomorphic if and only if the stabilisations of the graphs are orbit equivalent.

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.

A dynamical characterization of diagonal-preserving -isomorphisms of graph -algebras

2017 ◽  
Vol 38 (7) ◽  
pp. 2401-2421 ◽  
Author(s):  
SARA E. ARKLINT ◽  
SØREN EILERS ◽  
EFREN RUIZ
Keyword(s):  
Directed Graphs ◽  
Sufficient Condition ◽  
Orbit Equivalence ◽  
Necessary And Sufficient Condition ◽  
Graph Algebras ◽  
Boundary Path ◽  
Necessary And Sufficient

We characterize when there exists a diagonal-preserving $\ast$-isomorphism between two graph $C^{\ast }$-algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of ‘orbit equivalence’ between the boundary path spaces of the directed graphs $E$ and $F$ and show that this is a necessary and sufficient condition for the existence of a diagonal-preserving $\ast$-isomorphism between the graph $C^{\ast }$-algebras $C^{\ast }(E)$ and $C^{\ast }(F)$.

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.

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 RECOVERING THE BOUNDARY PATH SPACE OF A TOPOLOGICAL GRAPH USING POINTLESS TOPOLOGY

2020 ◽  
pp. 1-17
Author(s):  
GILLES G. DE CASTRO
Keyword(s):  
Path Space ◽  
Topological Graph ◽  
Topological Graphs ◽  
Locally Compact ◽  
C Algebra ◽  
Compact Spaces ◽  
Compact Topology ◽  
Boundary Path ◽  
Definition Of ◽  
Locally Compact Spaces

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_{1}$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.

scholarly journals Topological orbit equivalence classes and numeration scales of logistic maps

2011 ◽  
Vol 32 (5) ◽  
pp. 1501-1526
Author(s):  
MARÍA ISABEL CORTEZ ◽  
JUAN RIVERA-LETELIER
Keyword(s):  
Logistic Map ◽  
Natural Extension ◽  
Equivalence Classes ◽  
Orbit Equivalence ◽  
Orbit Equivalent ◽  
Uniquely Ergodic

AbstractWe show that every uniquely ergodic minimal Cantor system is topologically orbit equivalent to the natural extension of a numeration scale associated to a logistic map.

scholarly journals A short note on Cuntz splice from a viewpoint of continuous orbit equivalence of topological Markov shifts

2018 ◽  
Vol 123 (1) ◽  
pp. 91-100
Author(s):  
Kengo Matsumoto
Keyword(s):  
Short Note ◽  
Markov Shift ◽  
Orbit Equivalence ◽  
Irreducible Matrix ◽  
Orbit Equivalent ◽  
Markov Shifts ◽  
Topological Markov Shift ◽  
The One

Let $A$ be an $N\times N$ irreducible matrix with entries in $\{0,1\}$. We present an easy way to find an $(N+3)\times (N+3)$ irreducible matrix $\bar {A}$ with entries in $\{0,1\}$ such that the associated Cuntz-Krieger algebras ${\mathcal {O}}_A$ and ${\mathcal {O}}_{\bar {A}}$ are isomorphic and $\det (1 -A) = - \det (1-\bar {A})$. As a consequence, we find that two Cuntz-Krieger algebras ${\mathcal {O}}_A$ and ${\mathcal {O}}_B$ are isomorphic if and only if the one-sided topological Markov shift $(X_A, \sigma _A)$ is continuously orbit equivalent to either $(X_B, \sigma _B)$ or $(X_{\bar {B}}, \sigma _{\bar {B}})$.

scholarly journals Restricted orbit changes of ergodic ℤd-actions to achieve mixing and completely positive entropy

1986 ◽  
Vol 6 (4) ◽  
pp. 505-528 ◽  
Author(s):  
Adam Fieldsteel ◽  
N. A. Friedman
Keyword(s):  
Positive Entropy ◽  
Orbit Equivalence ◽  
Completely Positive ◽  
Orbit Equivalent

AbstractWe show that for every ergodic ℤd-action T, there is a mixing ℤd-action S which is orbit equivalent to T via an orbit equivalence that is a weak a-equivalence for all a ≥ 1 and a strong b-equivalence for all b ∈ (0, 1). If T has positive entropy, then S can be taken to have completely positive entropy. If the dimension d is greater than one, the orbit equivalence may be taken to be bounded and a strong b-equivalence for all b > 0.

scholarly journals Toeplitz flows and their ordered K-theory

2015 ◽  
Vol 36 (6) ◽  
pp. 1892-1921 ◽  
Author(s):  
SIRI-MALÉN HØYNES
Keyword(s):  
Analogous Result ◽  
Bratteli Diagram ◽  
Probability Measures ◽  
Orbit Equivalence ◽  
Incidence Matrices ◽  
Orbit Equivalent ◽  
Uncountable Family ◽  
Dimension Groups ◽  
Rank One ◽  
Formula Group

To a Toeplitz flow $(X,T)$ we associate an ordered $K^{0}$-group, denoted $K^{0}(X,T)$, which is order isomorphic to the $K^{0}$-group of the associated (non-commutative) $C^{\ast }$-crossed product $C(X)\rtimes _{T}\mathbb{Z}$. However, $K^{0}(X,T)$ can be defined in purely dynamical terms, and it turns out to be a complete invariant for (strong) orbit equivalence. We characterize the $K^{0}$-groups that arise from Toeplitz flows $(X,T)$ as exactly those simple dimension groups $(G,G^{+})$ that contain a non-cyclic subgroup $H$ of rank one that intersects $G^{+}$ non-trivially. Furthermore, the Bratteli diagram realization of $(G,G^{+})$ can be chosen to have the ERS property, i.e. the incidence matrices of the Bratteli diagram have equal row sums. We also prove that for any Choquet simplex $K$ there exists an uncountable family of pairwise non-orbit equivalent Toeplitz flows $(X,T)$ such that the set of $T$-invariant probability measures $M(X,T)$ is affinely homeomorphic to $K$, where the entropy $h(T)$ may be prescribed beforehand. Furthermore, the analogous result is true if we substitute strong orbit equivalence for orbit equivalence, but in that case we can actually prescribe both the entropy and the maximal equicontinuous factor of $(X,T)$. Finally, we present some interesting concrete examples of dimension groups associated to Toeplitz flows.

scholarly journals Continuous orbit equivalence rigidity

2016 ◽  
Vol 38 (4) ◽  
pp. 1543-1563 ◽  
Author(s):  
XIN LI
Keyword(s):  
Dynamical Systems ◽  
Classical Result ◽  
Crossed Products ◽  
Orbit Equivalence ◽  
Rigidity Results ◽  
Orbit Equivalent ◽  
Cartan Subalgebras ◽  
Duality Groups ◽  
Free Abelian Groups ◽  
Topological Dynamical Systems

We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterize continuous orbit equivalence in terms of isomorphisms of$C^{\ast }$-crossed products preserving Cartan subalgebras. This is the topological analogue of the classical result by Singer and Feldman-Moore in the measurable setting. Second, we turn to continuous orbit equivalence rigidity, i.e., the question whether for certain classes of topological dynamical systems, continuous orbit equivalence implies conjugacy. We show that this is not always the case by constructing topological dynamical systems (actions of free abelian groups and also non-abelian free groups) that are continuously orbit equivalent but not conjugate. Furthermore, we prove positive rigidity results. For instance, for solvable duality groups, general topological Bernoulli actions and certain subshifts of full shifts over finite alphabets are rigid.

Sign in / Sign up

Export Citation Format

Share Document