Stochastic adding machines based on Bratteli diagrams

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.

Download Full-text

Bratteli diagrams via the De Concini–Procesi theorem

Communications in Contemporary Mathematics ◽

10.1142/s021919972050073x ◽

2020 ◽

pp. 2050073

Author(s):

Daniele Mundici

Keyword(s):

Large Class ◽

Word Problem ◽

Bratteli Diagram ◽

Von Neumann ◽

Generators And Relations ◽

The Core ◽

Bratteli Diagrams ◽

Group Presentations ◽

Almost All ◽

Af Algebras

An AF algebra [Formula: see text] is said to be an AF[Formula: see text] algebra if the Murray–von Neumann order of its projections is a lattice. Many, if not most, of the interesting classes of AF algebras existing in the literature are AF[Formula: see text] algebras. We construct an algorithm which, on input a finite presentation (by generators and relations) of the Elliott semigroup of an AF[Formula: see text] algebra [Formula: see text], generates a Bratteli diagram of [Formula: see text] We generalize this result to the case when [Formula: see text] has an infinite presentation with a decidable word problem, in the sense of the classical theory of recursive group presentations. Applications are given to a large class of AF algebras, including almost all AF algebras whose Bratteli diagram is explicitly described in the literature. The core of our main algorithms is a combinatorial-polyhedral version of the De Concini–Procesi theorem on the elimination of points of indeterminacy in toric varieties.

Download Full-text

Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams

Acta Applicandae Mathematicae ◽

10.1007/s10440-018-0189-7 ◽

2018 ◽

Vol 159 (1) ◽

pp. 169-224 ◽

Cited By ~ 1

Author(s):

Sergey Bezuglyi ◽

Palle E. T. Jorgensen

Keyword(s):

Harmonic Functions ◽

Bratteli Diagrams

Download Full-text

Special triples from stationary Bratteli diagrams

The Michigan Mathematical Journal ◽

10.1307/mmj/1480734017 ◽

2016 ◽

Vol 65 (4) ◽

pp. 715-747 ◽

Cited By ~ 2

Author(s):

J. Kellendonk ◽

J. Savinien

Keyword(s):

Bratteli Diagrams

Download Full-text

Kneading sequences of strange adding machines

Topology and its Applications ◽

10.1016/j.topol.2008.11.018 ◽

2009 ◽

Vol 156 (17) ◽

pp. 2735-2746 ◽

Cited By ~ 4

Author(s):

Leslie Braziel Jones

Keyword(s):

Adding Machines

Download Full-text

Adding machines and wild attractors

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385797086392 ◽

1997 ◽

Vol 17 (6) ◽

pp. 1267-1287 ◽

Cited By ~ 12

Author(s):

HENK BRUIN ◽

GERHARD KELLER ◽

MATTHIAS ST. PIERRE

Keyword(s):

Dynamical System ◽

Critical Point ◽

Cantor Set ◽

Limit Set ◽

Unimodal Maps ◽

Irrational Rotations ◽

Circle Rotation ◽

Omega Limit Set ◽

Adding Machines

We investigate the dynamics of unimodal maps $f$ of the interval restricted to the omega limit set $X$ of the critical point for cases where $X$ is a Cantor set. In particular, many cases where $X$ is a measure attractor of $f$ are included. We give two classes of examples of such maps, both generalizing unimodal Fibonacci maps [LM, BKNS]. In all cases $f_{|X}$ is a continuous factor of a generalized odometer (an adding machine-like dynamical system), and at the same time $f_{|X}$ factors onto an irrational circle rotation. In some of the examples we obtain irrational rotations on more complicated groups as factors.

Download Full-text