Aperiodic substitution systems and their Bratteli diagrams

Bezuglyi, Sergey; Kwiatkowski, Jan; Medynets, Konstantin

doi:10.1017/s0143385708000230

Cited by 70 publications

(112 citation statements)

References 25 publications

(15 reference statements)

Supporting

Mentioning

109

Contrasting

Unclassified

Order By: Relevance

“…Moreover, the C * -algebras of higher-rank graphs are closely linked with orbit equivalence for shift spaces [10] and with symbolic dynamics more generally [43,47,44], as well as with fractals and self-similar structures [25,26]. More links between higher-rank graphs and symbolic dynamics can be seen via [3,4] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Monic representations of finite higher-rank graphs

Farsi¹,

Gillaspy²,

Jørgensen³

et al. 2018

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

In this paper we define the notion of monic representation for the C * -algebras of finite higher-rank graphs with no sources, and undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative C * -algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the Λ-semibranching representations previously studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal representation model for nonnegative monic representations.

show abstract

Section: Introductionmentioning

confidence: 99%

Monic representations of finite higher-rank graphs

Farsi¹,

Gillaspy²,

Jørgensen³

et al. 2018

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

“…In [DM08], T. Downarowicz and A. Maass proved that every Cantor minimal system of topological finite rank K > 1 is expansive. This result was generalized in [BKM09] to aperiodic Cantor dynamical systems of topological finite rank. Due to Hedlund [Hed69], every expansive Cantor dynamical system is conjugate to a subshift.…”

Section: Interpretation Of the Main Theorems In Terms Of Symbolic Dynmentioning

confidence: 93%

Exact number of ergodic invariant measures for Bratteli diagrams

Bezuglyi

Karpel

Kwiatkowski

2019

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

For a Bratteli diagram B, we study the simplex M1(B) of probability measures on the path space of B which are invariant with respect to the tail equivalence relation. Equivalently, M1(B) is formed by probability measures invariant with respect to a homeomorphism of a Cantor set. We study relations between the number of ergodic measures from M1(B) and the structure and properties of the diagram B. We prove a criterion and find sufficient conditions of unique ergodicity of a Bratteli diagram, in which case the simplex M1(B) is a singleton. For a finite rank k Bratteli diagram B having exactly l ≤ k ergodic invariant measures, we explicitly describe the structure of the diagram and find the subdiagrams which support these measures. We find sufficient conditions under which: (i) a Bratteli diagram has a prescribed number (finite or infinite) of ergodic invariant measures, and (ii) the extension of a measure from a uniquely ergodic subdiagram gives a finite ergodic invariant measure. Several examples, including stationary Bratteli diagrams, Pascal-Bratteli diagrams, and Toeplitz flows, are considered.

show abstract

“…To see this, we first notice that whenever d(λ) = (1, 1), R λ is an interval of the form (k/6, (k + 1)/6); there are six such paths λ, so each such interval is realized as R λ for some λ. 8 Indeed, for any such λ, one can calculate that τ λ is a linear function on a connected domain with slope −1/2. It follows that there are 12 paths λ with degree (2,2), and for each such λ we know that τ λ will be a linear function on a connected domain with slope 1/4.…”

Section: Indeed We Know Frommentioning

confidence: 99%

“…In addition to their relevance for C * -algebraic classification [78,86], the C * -algebras of higher-rank graphs are closely linked with orbit equivalence for shift spaces [17] and with symbolic dynamics more generally [79,88,80], with fractals and self-similar structures [35,36], and with renormalization problems in physics [40]. More links between higher-rank graphs and symbolic dynamics can be seen via [8,9,7] and the references cited therein. A wavelettype representation for higher-rank graphs was introduced in [37]; indeed, connections with wavelets, which had earlier been identified in certain special cases [12,30,67,31,75], were a major source of inspiration for the research we present below.…”

Section: Introductionmentioning

confidence: 99%

Separable representations, KMS states, and wavelets for higher-rank graphs

Farsi

Gillaspy

Kang

et al. 2016

Journal of Mathematical Analysis and Applications

124

View full text Add to dashboard Cite

Let Λ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of C * (Λ) on certain separable Hilbert spaces of the form L 2 (X, µ), by introducing the notion of a Λ-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, if Λ is aperiodic, we obtain a faithful representation ofwhere M is the Perron-Frobenius probability measure on the infinite path space Λ ∞ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a Λ-semibranching function system gives rise to KMS states for C * (Λ). For the higher-rank graphs of Robertson and Steger, we also obtain a representation of, where X is a fractal subspace of [0, 1] by embedding Λ ∞ into [0, 1] as a fractal subset X of [0,1]. In this latter case we additionally show that there exists a KMS state for C * (Λ) whose inverse temperature is equal to the Hausdorff dimension of X. Finally, we construct a wavelet system for L 2 (Λ ∞ , M ) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.2010 Mathematics Subject Classification: 46L05.

show abstract

Aperiodic substitution systems and their Bratteli diagrams

Cited by 70 publications

References 25 publications

Monic representations of finite higher-rank graphs

Monic representations of finite higher-rank graphs

Exact number of ergodic invariant measures for Bratteli diagrams

Separable representations, KMS states, and wavelets for higher-rank graphs

Contact Info

Product

Resources

About