Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
Research output: Contribution to journal › Journal article › Research › peer-review
Standard
Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids. / Carlsen, Toke Meier; Eilers, Soren; Ortega, Eduard; Restorff, Gunnar.
In: Journal of Mathematical Analysis and Applications, Vol. 469, No. 2, 2019, p. 1088-1110.Research output: Contribution to journal › Journal article › Research › peer-review
Harvard
APA
Vancouver
Author
Bibtex
}
RIS
TY - JOUR
T1 - Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
AU - Carlsen, Toke Meier
AU - Eilers, Soren
AU - Ortega, Eduard
AU - Restorff, Gunnar
PY - 2019
Y1 - 2019
N2 - We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph C*-algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz-Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy. (C) 2018 Elsevier Inc. All rights reserved.
AB - We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph C*-algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz-Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy. (C) 2018 Elsevier Inc. All rights reserved.
KW - Continuous orbit equivalence
KW - Flow equivalence
KW - Etale groupoids
KW - Graph C-algebras
KW - Leavitt path algebras
KW - Diagonal-preserving isomorphisms
U2 - 10.1016/j.jmaa.2018.09.056
DO - 10.1016/j.jmaa.2018.09.056
M3 - Journal article
VL - 469
SP - 1088
EP - 1110
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 2
ER -
ID: 209168491