Cantor Minimal Systems Research Articles

Extensions of Cantor minimal systems and dimension groups

Abstract. Given a factor map of Cantor minimal systems, we study the relations between the dimension groups of the two systems. First, we interpret the torsion subgroup of the quotient of the dimension groups in terms of intermediate extensions which are extensions of (Y,S) by a compact abelian group. Then we show that, by contrast, the existence of an intermediate non-abelian finite group extension can produce a situation where the dimension group of (Y,S) embeds into a proper subgroup of the dimension group of (X,T), yet the quotient of the dimension groups is nonetheless torsion free. Next we define higher order cohomology groups associated to an extension, and study them in various cases (proximal extensions, extensions by, not necessarily abelian, finite groups, etc.). Our main result here is that all the cohomology groups are torsion groups. As a consequence we can now identify as the torsion group of the quotient group .

Universally measure-preserving homeomorphisms of cantor minimal systems

In 1995, T. Giordano, I. Putnam, and C. Skau [GPS] made a significant breakthrough in Cantor minimal system theory. They completely classified Cantor minimal systems in the sense of topological orbit equivalence by using C*-algebra and homological algebra techniques. Since then, a dynamical proof of their theorem has been conjectured. Such a proof is presented in this paper. We establish orbit equivalence theory based on a finite coordinate change relation arising from an ordered Bratteli diagram, which is known from [HK] in the finitary case of ergodic probability measure-preserving transformations. We obtain the Orbital Extension Theorem. This theorem is considered a topological version of the Copying Lemma of Y. Katznelson and B. Weiss [KW], which has played an important role in measured orbit equivalence theory.

C*-algebras arising from substitutions

AbstractIn this paper, we introduce a C*-algebra associated with a primitive substitution. We show that when σ is proper, the C*-algebra is simple and purely infinite and contains the associated Cuntz–Krieger algebra and the crossed product C*-algebra of the corresponding Cantor minimal system. We calculate the K-groups.

On the eigenvalues of finite rank Bratteli–Vershik dynamical systems

AbstractIn this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated with the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition for there to be a measurable eigenvalue. Then, we consider two families of examples, a first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally, we study Toeplitz-type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove that measurable eigenvalues are always rational but not necessarily continuous.

Orbit equivalence of Cantor minimal systems: A survey and a new proof

We give a new proof of the classification, up to topological orbit equivalence, of minimal AF-equivalence relations and minimal actions of the group of integers on the Cantor set. This proof relies heavily on the structure of AF-equivalence relations and the theory of dimension groups; we give a short survey of these topics.

Not finitely but countably Hopf-equivalent clopen sets in a Cantor minimal system

We investigate topological Hopf-equivalences in the Morse substitution system. In particular, we give an example of not finitely but countably Hopf-equivalent clopen sets in a Cantor minimal system.

Induced subsystems associated to a Cantor minimal system

Induced subsystems associated to a Cantor minimal system

Finite-rank Bratteli–Vershik diagrams are expansive

AbstractThe representation of Cantor minimal systems by Bratteli–Vershik diagrams has been extensively used to study particular aspects of their dynamics. A main role has been played by the symbolic factors induced by the way vertices of a fixed level of the diagram are visited by the dynamics. The main result of this paper states that Cantor minimal systems that can be represented by Bratteli–Vershik diagrams with a uniformly bounded number of vertices at each level (called finite-rank systems) are either expansive or topologically conjugate to an odometer. More precisely, when expansive, they are topologically conjugate to one of their symbolic factors.

Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems

We consider the full group $[\varphi ]$ and topological full group $[[\varphi ]]$ of a Cantor minimal system $(X,\varphi )$. We prove that the commutator subgroups $D([\varphi ])$ and $D([[\varphi ]])$ are simple and show that the groups $D([\varphi ])$ a

On the subshift within a strong orbit equivalence class for minimal homeomorphisms

In this paper we will show that, for any strong orbit equivalence class for Cantor minimal systems, the class contains (minimal) subshifts which attain arbitrary finite topological entropy.

SOME REMARKS ON TOPOLOGICAL FULL GROUPS OF CANTOR MINIMAL SYSTEMS

Giordano, Putnam and Skau showed that topological full groups of Cantor minimal systems are complete invariants for flip conjugacy. We will completely determine the structure of normal subgroups of the topological full group. Moreover, a necessary and sufficient condition for the topological full group to be finitely generated will be given.

Approximate Conjugacy and Full Groups of Cantor Minimal Systems

H. Lin and the author introduced the notion of approximate conjugacy of dynamical systems. In this paper, we will discuss the relationship between approximate conjugacy and full groups of Cantor minimal systems. An analogue of Glasner–Weiss’s theorem will be shown. Approximate conjugacy of dynamical systems on the product of the Cantor set and the circle will also be studied.

TRACIAL EQUIVALENCE FOR $C^*$-ALGEBRAS AND ORBIT EQUIVALENCE FOR MINIMAL DYNAMICAL SYSTEMS

AbstractWe introduce the notion of tracial equivalence for $C^*$-algebras. Let $A$ and $B$ be two unital separable $C^*$-algebras. If they are tracially equivalent, then there are two sequences of asymptotically multiplicative contractive completely positive linear maps $\phi_n:A\to B$ and $\psi_n:B\to A$ with a tracial condition such that $\{\phi_n\circ\psi_n\}$ and $\{\psi_n\circ\phi_n\}$ are tracially approximately inner. Let $A$ and $B$ be two unital separable simple $C^*$-algebras with tracial topological rank zero. It is proved that $A$ and $B$ are tracially equivalent if and only if $A$ and $B$ have order isomorphic ranges of tracial states. For the Cantor minimal systems $(X_1,\sigma_1)$ and $(X_2,\sigma_2)$, using a result of Giordano, Putnam and Skau, we show that two such dynamical systems are (topological) orbit equivalent if and only if the associated crossed products $C(X_1)\times_{\sigma_1}\mathbb{Z}$ and $C(X_2)\times_{\sigma_2}\mathbb{Z}$ are tracially equivalent.

Minimal dynamical systems and approximate conjugacy

Several versions of approximate conjugacy for minimal dynamical systems are introduced. Relation between approximate conjugacy and corresponding crossed product C*-algebras is discussed. For the Cantor minimal systems, a complete description is given for these relations via K-theory and C*-algebras. For example, it is shown that two Cantor minimal systems are approximately τ-conjugate if and only if they are orbit equivalent and have the same periodic spectrum. It is also shown that two such systems are approximately K-conjugate if and only if the corresponding crossed product C*-algebras have the same scaled ordered K-theory. Consequently, two Cantor minimal systems are approximately K-conjugate if and only if the associated transformation C*-algebras are isomorphic. Incidentally, this approximate K-conjugacy coincides with Giordano, Putnam and Skau’s strong orbit equivalence for the Cantor minimal systems.

Affable equivalence relations and orbit structure of Cantor dynamical systems

We prove several new results about AF-equivalence relations and relate these to Cantor minimal systems (i.e. to minimal Z-actions). The results we obtain turn out to be crucial for the study of the topological orbit structure of more general countable group actions (as homeomorphisms) on Cantor sets, which will be the topic of a forthcoming paper. In all this, Bratteli diagrams and their dynamical interpretation are indispensable tools.

Dimension and infinitesimal groups of Cantor minimal systems

The dimension and infinitesimal groups of a Cantor dynamical system $(X,T)$ are inductive limits of sequences of homomorphisms defined by a proper Bratteli diagram of $(X,T)$. A method of selecting sequences of homomorphisms determining the dimension and the infinitesimal groups of $(X,T)$ based on non-proper Bratteli diagrams is described. The dimension and infinitesimal groups of Rudin-Shapiro, Morse and Chacon flows are computed.

Topological spectrum of locally compact Cantor minimal systems

We show that there exists a locally compact Cantor minimal system whose topological spectrum has a given Hausdor dimension.

SOME REMARKS ON TOPOLOGICAL ORBIT EQUIVALENCE OF CANTOR MINIMAL SYSTEMS

We will study the number of discontinuities of the orbit cocycles associated with orbit equivalence between Cantor minimal systems.

Topological orbit equivalence of locally compact Cantor minimal systems

Minimal homeomorphisms on the locally compact Cantor set are investigated. We prove that scaled dimension groups modulo infinitesimal subgroups determine topological orbit equivalence classes of locally compact Cantor minimal systems. We also introduce several full groups and show that they are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy. These are locally compact versions of the famous results for Cantor minimal systems obtained by Giordano et al. Moreover, proper homomorphisms and skew product extensions of locally compact Cantor minimal systems are examined and it is shown that every finite group can be embedded into the group of centralizers trivially acting on the dimension group.

Dimension groups of topological joinings and non-coalescence of Cantor minimal systems

When we have two extensions of a Cantor minimal system which are both one-to-one on at least one orbit, we can construct new Cantor minimal systems called topological joinings. We compute the dimension group of the joining in a special case. As an application, we show that a non-invertible endomorphism can induce the identity map on the dimension group of a Cantor minimal system.