scholarly journals Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz–Krieger algebras

2016 ◽  
Vol 285 (1-2) ◽  
pp. 121-141 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Orbit Equivalence ◽  
Circle Actions ◽  
Markov Shifts ◽  
Flow Equivalence

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 Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras

2014 ◽  
Vol 54 (4) ◽  
pp. 863-877 ◽  
Author(s):  
Kengo Matsumoto ◽  
Hiroki Matui
Keyword(s):  
Orbit Equivalence ◽  
Markov Shifts

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 On flow equivalence of one-sided topological Markov shifts

2016 ◽  
Vol 144 (7) ◽  
pp. 2923-2937 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Markov Shifts ◽  
Flow Equivalence

Orbit equivalence, flow equivalence and ordered cohomology

1996 ◽  
Vol 95 (1) ◽  
pp. 169-210 ◽  
Author(s):  
Mike Boyle ◽  
David Handelman
Keyword(s):  
Orbit Equivalence ◽  
Flow Equivalence

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 Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

2019 ◽  
Vol 469 (2) ◽  
pp. 1088-1110 ◽  
Author(s):  
Toke Meier Carlsen ◽  
Søren Eilers ◽  
Eduard Ortega ◽  
Gunnar Restorff
Keyword(s):  
Finite Type ◽  
Orbit Equivalence ◽  
Flow Equivalence
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