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 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 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 Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kengo Matsumoto
Keyword(s):  
Group Actions ◽  
Full Group ◽  
Orbit Equivalence ◽  
Cohomology Groups ◽  
Shift Spaces ◽  
Markov Shifts

<p style='text-indent:20px;'>We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.</p>

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 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 Cohomology groups invariant under continuous orbit equivalence

2020 ◽  
pp. 1-35
Author(s):  
Yongle Jiang
Keyword(s):  
Group Action ◽  
Continuous Group ◽  
Orbit Equivalence ◽  
Cohomology Groups ◽  
Bounded Cohomology ◽  
Homology Groups ◽  
Free Actions ◽  
Uniformly Finite Homology

By the work of Brodzki–Niblo–Nowak–Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variations of) these groups are invariants for topologically free actions under continuous orbit equivalence.

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