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

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 $C^{\ast }$-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 to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.

25 pages. Minor changes have been made and the list of references has been updated. This is the version that will be published

References in corpus (8)

• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
• The complete classification of unital graph $C^{\ast }$-algebras: Geometric and strong
• Geometric classification of graph $C^{\ast }$-algebras over finite graphs
• Diagonal-preserving graded isomorphisms of Steinberg algebras
• Orbit equivalence of graphs and isomorphism of graph groupoids
• $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$
• Flow equivalence and isotopy for subshifts
• A dynamical characterization of diagonal preserving $\ast$-isomorphisms of graph $C^{\ast }$-algebras

Cited by in corpus (15)

• Reconstruction of groupoids and C*-rigidity of dynamical systems
• Topological Full Groups of Ample Groupoids with Applications to Graph Algebras
• Orbit equivalence of graphs and isomorphism of graph groupoids
• Cuntz-Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids
• Conjugacy of local homeomorphisms via groupoids and C*-algebras
• C*-algebras, groupoids and covers of shift spaces
• Shift equivalences through the lens of Cuntz-Krieger algebras
• Subshift semigroups
• Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts
• One-sided topological conjugacy of topological Markov shifts, continuous full groups and Cuntz--Krieger algebras
• Orbit equivalence of higher-rank graphs
• Continuous orbit equivalence for automorphismhi systems of equivalence relations
• On one-sided topological conjugacy of topological Markov shifts and gauge actions on Cuntz--Krieger algebras
• $C^{\ast }$-rigidity of dynamical systems and étale groupoids
• Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz--Krieger algebras