Bratteli diagrams via the De Concini–Procesi theorem

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.

On measure preserving self-homeomorphisms of path spaces of simple stationary Bratteli diagrams

Borel measures which are invariant under the tail equivalence relation on path spaces of Bratteli diagrams are considered. We study the following problem: Let Bratteli diagram is fixed. Do every selfhomeomorphisms that preserve such a measure can be approximated by homeomorphisms which are “close to finitary” homeomorphisms? We found some conditions on diagrams for which it is true.

Bratteli–Vershik models for partial actions of ℤ

Let [Formula: see text] and [Formula: see text] be open subsets of the Cantor set with nonempty disjoint complements, and let [Formula: see text] be a homeomorphism with dense orbits. Building on the ideas of Herman, Putnam and Skau, we show that the partial action induced by [Formula: see text] can be realized as the Vershik map on an ordered Bratteli diagram, and that any two such diagrams are equivalent.

Finite-rank Bratteli-Vershik diagrams are expansive—a new proof

Downarowicz and Maass (Ergod. Th. and Dynam. Sys. 28 (2008), 739–747) proved that the Cantor minimal system associated to a properly ordered Bratteli diagram of finite rank is either an odometer system or an expansive system. We give a new proof of this truly remarkable result which we think is more transparent and easier to understand. We also address the question (Question 1) raised by Downarowicz and Maass and we find a better (i.e. lower) bound. In fact, we conjecture that the bound we have found is optimal.

Subdiagrams and invariant measures on Bratteli diagrams

We study ergodic finite and infinite measures defined on the path space $X_{B}$ of a Bratteli diagram $B$ which are invariant with respect to the tail equivalence relation on $X_{B}$. Our interest is focused on measures supported by vertex and edge subdiagrams of $B$. We give several criteria when a finite invariant measure defined on the path space of a subdiagram of $B$ extends to a finite invariant measure on $B$. Given a finite ergodic measure on a Bratteli diagram $B$ and a subdiagram $B^{\prime }$ of $B$, we find the necessary and sufficient conditions under which the measure of the path space $X_{B^{\prime }}$ of $B^{\prime }$ is positive. For a class of Bratteli diagrams of finite rank, we determine when they have maximal possible number of ergodic invariant measures. The case of diagrams of rank two is completely studied. We also include an example which explicitly illustrates the proven results.

Toeplitz flows and their ordered K-theory

To a Toeplitz flow $(X,T)$ we associate an ordered $K^{0}$-group, denoted $K^{0}(X,T)$, which is order isomorphic to the $K^{0}$-group of the associated (non-commutative) $C^{\ast }$-crossed product $C(X)\rtimes _{T}\mathbb{Z}$. However, $K^{0}(X,T)$ can be defined in purely dynamical terms, and it turns out to be a complete invariant for (strong) orbit equivalence. We characterize the $K^{0}$-groups that arise from Toeplitz flows $(X,T)$ as exactly those simple dimension groups $(G,G^{+})$ that contain a non-cyclic subgroup $H$ of rank one that intersects $G^{+}$ non-trivially. Furthermore, the Bratteli diagram realization of $(G,G^{+})$ can be chosen to have the ERS property, i.e. the incidence matrices of the Bratteli diagram have equal row sums. We also prove that for any Choquet simplex $K$ there exists an uncountable family of pairwise non-orbit equivalent Toeplitz flows $(X,T)$ such that the set of $T$-invariant probability measures $M(X,T)$ is affinely homeomorphic to $K$, where the entropy $h(T)$ may be prescribed beforehand. Furthermore, the analogous result is true if we substitute strong orbit equivalence for orbit equivalence, but in that case we can actually prescribe both the entropy and the maximal equicontinuous factor of $(X,T)$. Finally, we present some interesting concrete examples of dimension groups associated to Toeplitz flows.

Perfect Orderings on Finite Rank Bratteli Diagrams

Given a Bratteli diagram B, we study the set 𝒪B of all possible orderings on B and its subset PB consisting of perfect orderings that produce Bratteli–Vershik topological dynamical systems (Vershik maps). We give necessary and sufficient conditions for the ordering ω to be perfect. On the other hand, a wide class of non-simple Bratteli diagrams that do not admit Vershik maps is explicitly described. In the case of finite rank Bratteli diagrams, we show that the existence of perfect orderings with a prescribed number of extreme paths constrains significantly the values of the entries of the incidence matrices and the structure of the diagram B. Our proofs are based on the new notions of skeletons and associated graphs, defined and studied in the paper. For a Bratteli diagram B of rank k, we endow the set 𝒪B with product measure μ and prove that there is some 1 ≤ j ≤ k such that μ-almost all orderings on B have j maximal and j minimal paths. If j is strictly greater than the number of minimal components that B has, then μ-almost all orderings are imperfect.

On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type

Product type equivalence relations are hyperfinitemeasured equivalence relations, which, up to orbit equivalence, are generated by product type odometer actions. We give a concrete example of a hyperfinite equivalence relation of non-product type, which is the tail equivalence on a Bratteli diagram. In order to show that the equivalence relation constructed is not of product type we will use a criterion called property A. This property, introduced by Krieger for non-singular transformations, is defined directly for hyperfinite equivalence relations in this paper.

On the eigenvalues of finite rank Bratteli–Vershik dynamical systems

In 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.

Invariant measures on stationary Bratteli diagrams

We study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we give an explicit description of all ergodic probability measures that are invariant with respect to the tail equivalence relation (or the Vershik map); these measures are completely described by the incidence matrix of the Bratteli diagram. Since such diagrams correspond to substitution dynamical systems, our description provides an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.