Cantor Minimal Systems Research Articles

Bratteli-Vershik models for Cantor minimal systems associated to interval exchange transformations

We construct Bratteli-Vershik models for minimal interval exchange transformations. We use this to show that the interval exchange transformations over quadratic fields, recently studied by Boshernitzan and Carroll, actually are (conjugate to) substitution minimal systems. We also prove a partial converse to this. Furthermore, these systems are orbit equivalent to Sturmian systems.

Finite order automorphisms and dimension groups of Cantor minimal systems

We compute the dimension group of the skew product extension of a Cantor minimal system associated with a finite group valued cocycle. Using it, we study finite subgroups in the commutant group of a Cantor minimal system and prove that a finite subgroup of the kernel of the mod map must be cyclic. Moreover, we give a certain obstruction for finite subgroups of commutant groups to have nonzero intersection to the kernel of mod maps. We also give a necessary and sufficient condition for dimension groups so that the kernel of the mod map can include a finite order element.

Toeplitz flows, ordered Bratteli diagrams and strong orbit equivalence

Let (X,T) be a Cantor minimal system associated with a properly ordered Bratteli diagram with the equal path number property. In this paper we will show that there exists a Toeplitz flow (Y,S) such that (X,T) and (Y,S) are strongly orbit equivalent. This is an affirmative answer to the open problem in R. Gjerde and Ø. Johansen (Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows. Ergod. Th. & Dynam. Sys.20 (2000), 1687–1710).

Limit laws of entrance timesfor low-complexity Cantor minimal systems

This paper is devoted to the study oflimit laws of entrance times to cylindersets for Cantor minimal systems of zeroentropy using their representation bymeans of ordered Bratteli diagrams. Westudy in detail substitution subshifts andwe prove that these limit laws arepiecewise-linear functions. The same kindof results are obtained for classicallow-complexity systems given bynon-stationary ordered Bratteli diagrams.

Ext and OrderExt Classes of Certain Automorphisms of C*-Algebras Arising from Cantor Minimal Systems

AbstractGiordano, Putnam and Skau showed that the transformation group C*-algebra arising from a Cantor minimal system is an AT-algebra, and classified it by its K-theory. For approximately inner automorphisms that preserve C(X), we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto’s result.

The topological full group of a Cantor minimal system is dense in the full group

To every homeomorphism $T$ of a Cantor set $X$ one can associate the full group $[T]$ formed by all homeomorphisms $\gamma$ such that $\gamma(x)=T^{n(x)}(x)$, $ x\in X$. The topological full group $[[T]]$ consists of all homeomorphisms whose associated orbit cocycle $n(x)$ is continuous. The uniform and weak topologies, $\tau_u$ and $\tau_w$, as well as their intersection $\tau_{uw}$ are studied on $\text{\rm Homeo}(X)$. It is proved that $[[T]]$ is dense in $[T]$ with respect to $\tau_u$. A Cantor minimal system $(X,T)$ is called saturated if any two clopen sets of ``the same measure'' are $[[T]]$-equivalent. We describe the class of saturated Cantor minimal systems. In particular, $(X,T)$ is saturated if and only if the closure of $[[T]]$ in $\tau_{uw}$ is $[T]$ and if and only if every infinitesimal function is a $T$-coboundary. These results are based on a description of homeomorphisms from $[[T]]$ related to a given sequence of Kakutani-Rokhlin partitions. It is shown that the offered method works for some symbolic Cantor minimal systems. The tool of Kakutani-Rokhlin partitions is used to characterize $[[T]]$-equivalent clopen sets and the subgroup $[[T]]_x \subset [[T]]$ formed by homeomorphisms preserving the forward orbit of $x$.

Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows

We construct Bratteli–Vershik models for Toeplitz flows and characterize a class of properly ordered Bratteli diagrams corresponding to these flows. We use this result to extend by a novel approach—using basic theory of dimension groups—an interesting and non-trivial result about Toeplitz flows, first shown by Downarowicz. (Williams had previously obtained preliminary results in this direction.) The result states that to any Choquet simplex $K$, there exists a $0$–$1$ Toeplitz flow $(Y,\psi)$, so that the set of invariant probability measures of $(Y,\psi)$ is affinely homeomorphic to $K$. Not only do we give a conceptually new proof of this result, we also show that we may choose $(Y,\psi)$ to have zero entropy and to have full rational spectrum.Furthermore, our Bratteli–Vershik model for a given Toeplitz flow explicitly exhibits the factor map onto the maximal equicontinuous (odometer) factor. We utilize this to give a simple proof of the existence of a uniquely ergodic 0–1 Toeplitz flow of zero entropy having a given odometer as its maximal equicontinuous factor and being strongly orbit equivalent to this factor. By the same token, we show the existence of 0–1 Toeplitz flows having the 2-odometer as their maximal equicontinuous factor, being strong orbit equivalent to the same, and assuming any entropy value in $[0,\ln 2)$.Finally, we show by an explicit example, using Bratteli diagrams, that Toeplitz flows are not preserved under Kakutani equivalence (in fact, under inducing)—contrasting what is the case for substitution minimal systems. In fact, the example we exhibit is an induced system of a 0–1 Toeplitz flow which is conjugate to the Chacon substitution system, thus it is prime, i.e. it has no non-trivial factors.The thrust of our paper is to demonstrate the relevance and usefulness of Bratteli–Vershik models and dimension group theory for the study of minimal symbolic systems. This is also exemplified in recent papers by Forrest and by Durand, Host and Skau, treating substitution minimal systems, and by papers by Boyle, Handelman and by Ormes.

Full groups of Cantor minimal systems

We associate different types of full groups to Cantor minimal systems. We show how these various groups (as abstract groups) are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy, respectively. Furthermore, we introduce a group homomorphism, the socalled mod map, from the normalizers of the various full groups to the automorphism groups of the (ordered)K 0-groups, which are associated to the Cantor minimal systems. We show how this in turn is related to the automorphisms of the associatedC *-crossed products. Our results are analogues in the topological dynamical setting of results obtained by Dye, Connes-Krieger and Hamachi-Osikawa in measurable dynamics.

WEAK ORBIT EQUIVALENCE OF CANTOR MINIMAL SYSTEMS

This paper is a commentary on the recent work [4]. It has two goals: the first is to eliminate the C*-algebra machinery from the proofs of the results of [4]; the second, to provide a characterization of weak orbit equivalence of Cantor minimal systems in terms of their dimension groups.