Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids |
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Authors: | Toke Meier Carlsen Søren Eilers Eduard Ortega Gunnar Restorff |
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Affiliation: | 1. Department of Science and Technology, University of the Faroe Islands, Nóatún 3, FO-100 Tórshavn, the Faroe Islands;2. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark;3. Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway |
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Abstract: | 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 -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. |
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Keywords: | Continuous orbit equivalence Flow equivalence Étale groupoids Leavitt path algebras Diagonal-preserving isomorphisms |
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