Continuous Orbit Equivalence Research Articles

Continuous Orbit Equivalence for Automorphism Systems of Equivalence Relations

Continuous Orbit Equivalence for Automorphism Systems of Equivalence Relations

On $\mathbb{Z}^{d}$-odometers associated to integer matrices

We extend the results by Giordano, Putnam, and Skau (2019) on characterization of conjugacy, isomorphism, and continuous orbit equivalence of \mathbb{Z}^{d} -odometers to dimensions d>2 . We then apply these extensions to the case of odometers defined by matrices with integer coefficients.

Continuous Orbit Equivalence of Semigroup Actions

Continuous Orbit Equivalence of Semigroup Actions

Finitely presented isomorphisms of Cuntz-Krieger algebras and continuous orbit equivalence of one-sided topological Markov shifts

We introduce the notion of finitely presented isomorphism between Cuntz–Krieger algebras, and of finitely presented isomorphic Cuntz–Krieger algebras. We prove that there exists a finitely presented isomorphism between Cuntz–Krieger algebras $\mathcal{O}_A$ and $\mathcal{O}_B$ if and only if their one-sided topological Markov shifts $(X_A,\sigma_A)$ and $(X_B,\sigma_B)$ are continuously orbit equivalent. Hence the value $\det (I-A)$ is a complete invariant for the existence of a finitely presented isomorphism between isomorphic Cuntz–Krieger algebras, so that there exists a pair of Cuntz–Krieger algebras which are isomorphic but not finitely presented isomorphic.

Corrigendum to “Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz–Krieger algebras”

In the given proofs of Theorem 1.5 in the author’s article "K. Matsumoto, Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz–Krieger algebras", Proc. Amer. Math. Soc. 145 (2017), pp. 1131–1140, there are two places that the matrices are needed to be primitive. The statement of Theorem 1.5 and its proof are corrected.

Continuous orbit equivalence rigidity for left-right wreath product actions

Drimbe and Vaes proved an orbit equivalence superrigidity theorem for left-right wreath product actions in the measurable setting. We establish the counterpart result in the topological setting for continuous orbit equivalence. This gives us minimal, topologically free actions that are continuous orbit equivalence superrigid. One main ingredient for the proof is to show continuous cocycle superrigidity for certain generalized full shifts, extending our previous result with Chung.

Subgroups of continuous full groups and relative continuous orbit equivalences of one-sided topological Markov shifts

Let A be an irreducible non-permutation matrix with entries in {0,1}. We study a family ΓA,f of subgroups of the continuous full group ΓA of the one-sided topological Markov shift (XA,σA). The subgroup ΓA,f is indexed by an integer valued continuous function f on XA such that ΓA,0=ΓA the non-amenable continuous full group and ΓA,1=ΓAAF the amenable AF full group. We will first prove that a spatial realization theorem for the subgroups ΓA,f under certain conditions on the functions f. We will second introduce relative versions of continuous orbit equivalence to study classification of the subgroups ΓA,f related to their associated groupoids and C⁎-algebras.

On continuous orbit equivalence rigidity for virtually cyclic group actions

We prove that any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group by certain other virtually cyclic groups, e.g., the direct product of the integer group with any non-abelian finite simple group.

KMS states and continuous orbit equivalence for ultragraph shift spaces with sinks

We extend ultragraph shift spaces and the realization of ultragraph C*-algebras as partial crossed products to include ultragraphs with sinks (under a mild condition, called (RFUM2), which allow us to dismiss the use of filters) and we describe the associated transformation groupoid. Using these characterizations we study continuous orbit equivalence of ultragraph shift spaces (via groupoids) and KMS and ground states (via partial crossed products).

Cohomology groups, continuous full groups and continuous orbit equivalence of topological 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>

Cantor dynamics of renormalizable groups

A group \Gamma is said to be “finitely non-co-Hopfian,” or “renormalizable,” if there exists a self-embedding \varphi \colon \Gamma \to \Gamma whose image is a proper subgroup of finite index. Such a proper self-embedding is called a “renormalization for \Gamma .” In this work, we associate a dynamical system to a renormalization \varphi of \Gamma . The discriminant invariant {\mathcal D}_{\varphi} of the associated Cantor dynamical system is a profinite group which is a measure of the asymmetries of the dynamical system. If {\mathcal D}_{\varphi} is a finite group for some renormalization, we show that \Gamma/C_{\varphi} is virtually nilpotent, where C_{\varphi} is the kernel of the action map. We introduce the notion of a (virtually) renormalizable Cantor action, and show that the action associated to a renormalizable group is virtually renormalizable. We study the properties of virtually renormalizable Cantor actions, and show that virtual renormalizability is an invariant of continuous orbit equivalence. Moreover, the discriminant invariant of a renormalizable Cantor action is an invariant of continuous orbit equivalence. Finally, the notion of a renormalizable Cantor action is related to the notion of a self-replicating group of automorphisms of a rooted tree.

Orbit equivalence of higher-rank graphs

We study the notion of continuous orbit equivalence of finitely-aligned higher-rank graphs. We show that there is a continuous orbit equivalence between two finitely-aligned higher-rank graphs that preserves the periodicity of boundary paths if and only if the boundary path groupoids are isomorphic. We also study eventual one-sided conjugacy of finitely-aligned higher-rank graphs and two-sided conjugacy of row-finite higher-rank graphs.

Nilpotent Cantor actions

A nilpotent Cantor action is a minimal equicontinuous action $\Phi \colon \Gamma \times \frak{X} \to \frak{X}$ on a Cantor set $\frak{X}$, where $\Gamma$ contains a finitely-generated nilpotent subgroup $\Gamma_0 \subset \Gamma$ of finite index. In this note, we show that these actions are distinguished among general Cantor actions: any effective action of a finitely generated group on a Cantor space, which is continuously orbit equivalent to a nilpotent Cantor action, must itself be a nilpotent Cantor action. As an application of this result, we obtain new invariants of nilpotent Cantor actions under continuous orbit equivalence.

Asymptotic continuous orbit equivalence of expansive systems

We introduce notions of asymptotic continuous orbit equivalence and (strongly) asymptotic conjugacy for expansive systems, and characterize them in terms of the transformation groupoids, the principal groupoids coming from the local conjugacy relations an

Coded equivalence of one-sided topological Markov shifts

We introduce a notion of coded equivalence in one-sided topological Markov shifts. The notion is inspired by coding theory. One-sided topological conjugacy implies coded equivalence. We will show that coded equivalence implies continuous orbit equivalence of one-sided topological Markov shifts.

𝐶*-algebras, groupoids and covers of shift spaces

To every one-sided shift space X \mathsf {X} we associate a cover X ~ \widetilde {\mathsf {X}} , a groupoid G X \mathcal {G}_\mathsf {X} and a C ∗ \mathrm {C^*} -algebra O X \mathcal {O}_\mathsf {X} . We characterize one-sided conjugacy, eventual conjugacy and (stabilizer-preserving) continuous orbit equivalence between X \mathsf {X} and Y \mathsf {Y} in terms of isomorphism of G X \mathcal {G}_\mathsf {X} and G Y \mathcal {G}_\mathsf {Y} , and diagonal-preserving ∗ ^* -isomorphism of O X \mathcal {O}_\mathsf {X} and O Y \mathcal {O}_\mathsf {Y} . We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces Λ X \Lambda _\mathsf {X} and Λ Y \Lambda _\mathsf {Y} in terms of isomorphism of the stabilized groupoids G X × R \mathcal {G}_\mathsf {X}\times \mathcal {R} and G Y × R \mathcal {G}_\mathsf {Y}\times \mathcal {R} , and diagonal-preserving ∗ ^* -isomorphism of the stabilized C ∗ \mathrm {C^*} -algebras O X ⊗ K \mathcal {O}_\mathsf {X}\otimes \mathbb {K} and O Y ⊗ K \mathcal {O}_\mathsf {Y}\otimes \mathbb {K} . Our strategy is to lift relations on the shift spaces to similar relations on the covers. Restricting to the class of sofic shifts whose groupoids are effective, we show that it is possible to recover the continuous orbit equivalence class of X \mathsf {X} from the pair ( O X , C ( X ) ) (\mathcal {O}_\mathsf {X}, C(\mathsf {X})) , and the flow equivalence class of Λ X \Lambda _\mathsf {X} from the pair ( O X ⊗ K , C ( X ) ⊗ c 0 ) (\mathcal {O}_\mathsf {X}\otimes \mathbb {K}, C(\mathsf {X})\otimes c_0) . In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.

Limit group invariants for non-free Cantor actions

A Cantor action is a minimal equicontinuous action of a countably generated group$G$on a Cantor space$X$. Such actions are also called generalized odometers in the literature. In this work, we introduce two new conjugacy invariants for Cantor actions, the stabilizer limit group and the centralizer limit group. An action is wild if the stabilizer limit group is an increasing sequence of stabilizer groups without bound and otherwise is said to be stable if this group chain is bounded. For Cantor actions by a finitely generated group$G$, we prove that stable actions satisfy a rigidity principle and furthermore show that the wild property is an invariant of the continuous orbit equivalence class of the action. A Cantor action is said to be dynamically wild if it is wild and the centralizer limit group is a proper subgroup of the stabilizer limit group. This property is also a conjugacy invariant and we show that a Cantor action with a non-Hausdorff element must be dynamically wild. We then give examples of wild Cantor actions with non-Hausdorff elements, using recursive methods from geometric group theory to define actions on the boundaries of trees.

A groupoid approach to C*-algebras associated with λ-graph systems and continuous orbit equivalence of subshifts

ABSTRACT A λ-graph system is a labelled Bratteli diagram with shift operation. It is a generalized notion of finite labelled graph and presents a subshift. We will study continuous orbit equivalence of one-sided subshifts and topological conjugacy of two-sided subshifts from the view points of groupoids and -algebras constructed from λ-graph systems.

Cohomology groups invariant under continuous orbit equivalence

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.

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

We study KMS states for gauge actions with potential functions on Cuntz--Krieger algebras whose underlying one-sided topological Markov shifts are continuous orbit equivalent. As a result, we have a certain relationship between topological entropy of continuous orbit equivalent one-sided topological Markov shifts.