Cantor Minimal Systems

2018 ◽  
Author(s):  
Ian Putnam
Keyword(s):  
Cantor Minimal Systems

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

2000 ◽  
Vol 20 (6) ◽  
pp. 1687-1710 ◽  
Author(s):  
RICHARD GJERDE ◽  
ØRJAN JOHANSEN
Keyword(s):  
Dimension Group ◽  
Orbit Equivalent ◽  
Bratteli Diagrams ◽  
Novel Approach ◽  
Dimension Groups ◽  
Substitution System ◽  
Zero Entropy ◽  
Factor Map ◽  
Cantor Minimal Systems ◽  
Uniquely Ergodic

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.

WEAK ORBIT EQUIVALENCE OF CANTOR MINIMAL SYSTEMS

1995 ◽  
Vol 06 (04) ◽  
pp. 559-579 ◽  
Author(s):  
ELI GLASNER ◽  
BENJAMIN WEISS
Keyword(s):  
Recent Work ◽  
Orbit Equivalence ◽  
C Algebra ◽  
Dimension Groups ◽  
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.

Full groups of Cantor minimal systems

1999 ◽  
Vol 111 (1) ◽  
pp. 285-320 ◽  
Author(s):  
Thierry Giordano ◽  
Ian F. Putnam ◽  
Christian F. Skau
Keyword(s):  
Cantor Minimal Systems

scholarly journals Dimension and infinitesimal groups of Cantor minimal systems

2004 ◽  
Vol 23 (1) ◽  
pp. 161 ◽  
Author(s):  
Jan Kwiatkowski ◽  
Marcin Wata
Keyword(s):  
Cantor Minimal Systems

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

2001 ◽  
Vol 53 (2) ◽  
pp. 325-354 ◽  
Author(s):  
Hiroki Matui
Keyword(s):  
Transformation Group ◽  
Minimal System ◽  
Full Group ◽  
C Algebras ◽  
Inner Automorphism ◽  
Inner Automorphisms ◽  
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.

Finite-rank Bratteli–Vershik diagrams are expansive

2008 ◽  
Vol 28 (3) ◽  
pp. 739-747 ◽  
Author(s):  
TOMASZ DOWNAROWICZ ◽  
ALEJANDRO MAASS
Keyword(s):  
Finite Rank ◽  
Main Role ◽  
Fixed Level ◽  
Bounded Number ◽  
Cantor Minimal Systems ◽  
Uniformly Bounded ◽  
The Way

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.

scholarly journals On the eigenvalues of finite rank Bratteli–Vershik dynamical systems

2009 ◽  
Vol 30 (3) ◽  
pp. 639-664 ◽  
Author(s):  
XAVIER BRESSAUD ◽  
FABIEN DURAND ◽  
ALEJANDRO MAASS
Keyword(s):  
Finite Rank ◽  
Additive Group ◽  
Bratteli Diagram ◽  
Necessary Condition ◽  
Incidence Matrices ◽  
Bounded Number ◽  
Stable Space ◽  
To Come ◽  
Cantor Minimal Systems ◽  
Stable Subspace

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.

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