Asymptotic -Algebras from -Actions on Higher Rank Graphs

Abstract and Applied Analysis ◽

10.1155/2013/752497 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5

Author(s):

Inhyeop Yi

Keyword(s):

Dynamical System ◽

Semidirect Product ◽

Equivalence Relation ◽

Asymptotic Equivalence ◽

Vertex Set ◽

Higher Rank

For a dynamical system arising from -action on a higher rank graph with finite vertex set, we show that the semidirect product of the asymptotic equivalence relation groupoid is essentially principal if and only if the -graph satisfies the aperiodic condition. Then we show that the corresponding asymptotic Ruelle algebra is simple if the graph is primitive with the aperiodic condition.

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OBSTRUCTIONS TO A GENERAL CHARACTERIZATION OF GRAPH CORRESPONDENCES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788713000177 ◽

2013 ◽

Vol 95 (2) ◽

pp. 169-188

Author(s):

S. KALISZEWSKI ◽

NURA PATANI ◽

JOHN QUIGG

Keyword(s):

Directed Graph ◽

Discrete Space ◽

Topological Graphs ◽

Locally Compact ◽

General Characterization ◽

Product Systems ◽

Vertex Set ◽

Higher Rank

AbstractFor a countable discrete space $V$, every nondegenerate separable ${C}^{\ast } $-correspondence over ${c}_{0} (V)$ is isomorphic to one coming from a directed graph with vertex set $V$. In this paper we demonstrate why the analogous characterizations fail to hold for higher-rank graphs (where one considers product systems of ${C}^{\ast } $-correspondences) and for topological graphs (where $V$ is locally compact Hausdorff), and we discuss the obstructions that arise.

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Graph Labelings and Continuous Factors of Dynamical Systems

The Electronic Journal of Combinatorics ◽

10.37236/2213 ◽

2013 ◽

Vol 20 (1) ◽

Author(s):

Stephen M. Shea

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Topological Entropy ◽

Directed Graphs ◽

Sufficient Conditions ◽

Perfect Graph ◽

Shift Space ◽

Finite Set ◽

Vertex Set ◽

Definition Of

A labeling of a graph is a function from the vertex set of the graph to some finite set. Certain dynamical systems (such as topological Markov shifts) can be defined by directed graphs. In these instances, a labeling of the graph defines a continuous, shift-commuting factor of the dynamical system. We find sufficient conditions on the labeling to imply classification results for the factor dynamical system. We define the topological entropy of a (directed or undirected) graph and its labelings in a way that is analogous to the definition of topological entropy for a shift space in symbolic dynamics. We show, for example, if $G$ is a perfect graph, all proper $\chi(G)$-colorings of $G$ have the same entropy, where $\chi(G)$ is the chromatic number of $G$.

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A note on the ideals of groupoid C*-algebras from Smale spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700039939 ◽

2001 ◽

Vol 64 (2) ◽

pp. 271-279 ◽

Cited By ~ 1

Author(s):

Chengjun Hou ◽

Xiamoman Chen

Keyword(s):

Equivalence Relation ◽

Asymptotic Equivalence ◽

Topologically Transitive ◽

C Algebras ◽

Smale Space ◽

Smale Spaces

In this note, we characterise completely the ideals of the groupoid C*-algebra arising from the asymptotic equivalence relation on the points of a Smale space and show that the related Ruelle algebra is simple when the Smale space is topologically transitive.

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A generalization of the simulation theorem for semidirect products

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.21 ◽

2018 ◽

Vol 39 (12) ◽

pp. 3185-3206

Author(s):

SEBASTIÁN BARBIERI ◽

MATHIEU SABLIK

Keyword(s):

Dynamical System ◽

Finite Type ◽

Word Problem ◽

Semidirect Product ◽

Finitely Generated ◽

Subshift Of Finite Type ◽

Semidirect Products ◽

Finitely Generated Group

We generalize a result of Hochman in two simultaneous directions: instead of realizing an arbitrary effectively closed $\mathbb{Z}^{d}$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^{2}$. Let $H$ be a finitely generated group and $G=\mathbb{Z}^{2}\rtimes _{\unicode[STIX]{x1D711}}H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,T)$ where $Y\subset \{0,1\}^{\mathbb{N}}$, there exists a $G$-subshift of finite type $(X,\unicode[STIX]{x1D70E})$ such that the $H$-subaction of $(X,\unicode[STIX]{x1D70E})$ is an extension of $(Y,T)$. In the case where $T$ is an expansive action, a subshift conjugated to $(Y,T)$ can be obtained as the $H$-projective subdynamics of a sofic $G$-subshift. As a corollary, we obtain that $G$ admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of $H$ is decidable.

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Subrelations of ergodic equivalence relations

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700004958 ◽

1989 ◽

Vol 9 (2) ◽

pp. 239-269 ◽

Cited By ~ 23

Author(s):

J. Feldman ◽

C. E. Sutherland ◽

R. J. Zimmer

Keyword(s):

Finite Index ◽

Lie Group ◽

Equivalence Relation ◽

Equivalence Relations ◽

Type Ii ◽

Compact Lie Group ◽

Rigidity Results ◽

Higher Rank ◽

Ergodic Equivalence Relations ◽

Inner Automorphisms

AbstractWe introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II1. Such pairs are characterized by a groupQwhich serves as a quotient for the pair, or by the ability to synthesize the larger relation from the smaller and an action (modulo inner automorphisms) ofQon it; in the case whereQis amenable, one can work with a genuine action. We classify ergodic subrelations of finite index, and arbitrary normal subrelations, of the unique amenable relation of type II1. We also give a number of rigidity results; for example, if an equivalence relation is generated by a free II1-action of a lattice in a higher rank simple connected non-compact Lie group with finite centre, the only normal ergodic subrelations are of finite index, and the only strongly normal, amenable subrelations are finite.

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Direct and semidirect product of n-ary polygroups via n-ary factor polygroups

Journal of Algebra and Its Applications ◽

10.1142/s0219498819500828 ◽

2019 ◽

Vol 18 (05) ◽

pp. 1950082 ◽

Cited By ~ 1

Author(s):

Lumnije Shehu ◽

Bijan Davvaz

Keyword(s):

Direct Product ◽

Semidirect Product ◽

Equivalence Relation

In this paper, we define an equivalence relation induced by [Formula: see text]-ary subpolygroups and show that such relation is full conjugation when [Formula: see text]-ary subpolygroup is normal. Then, we introduce a construction and some properties of direct product of [Formula: see text]-ary polygroups via [Formula: see text]-ary factor polygroups. This construction in principle is enlargement of elements of [Formula: see text]-ary polygroup except zero. Finally, we introduce and investigate the semidirect product of [Formula: see text]-ary polygroups via [Formula: see text]-ary factor polygroups.

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The weak and the strong equivalence relation and the asymptotic inversion

Filomat ◽

10.2298/fil1104029d ◽

2011 ◽

Vol 25 (4) ◽

pp. 29-36

Author(s):

Dragan Djurcic ◽

Ivan Mitrovic ◽

Mladen Janjic

Keyword(s):

Equivalence Relation ◽

Functional Class ◽

Asymptotic Equivalence ◽

Measurable Functions ◽

Functional Classes ◽

Half Axis ◽

The Relationship ◽

Rapidly Varying Functions

In this paper we discuss the relationship between the weak and the strong asymptotic equivalence relation and the asymptotic inversion, for positive and measurable functions defined on a half-axis [a,+?) (a > 0). As the main results, we prove a certain characterizations of the functional class of all rapidly varying functions, as well as some other functional classes.

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Simple dynamical systems

Applied General Topology ◽

10.4995/agt.2019.7910 ◽

2019 ◽

Vol 20 (2) ◽

pp. 307 ◽

Cited By ~ 1

Author(s):

K. Ali Akbar ◽

V. Kannan ◽

I. Subramania Pillai

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Equivalence Class ◽

Equivalence Relation ◽

Title Page ◽

The Family ◽

Dynamical Properties ◽

Simple Systems

<div class="page" title="Page 1"><div class="layoutArea"><div class="column">In this paper, we study the class of simple systems on R induced by homeomorphisms having finitely many non-ordinary points. We characterize the family of homeomorphisms on R having finitely many non-ordinary points upto (order) conjugacy. For x,y ∈ R, we say x ∼ y on a dynamical system (R,f) if x and y have same dynamical properties, which is an equivalence relation. Said precisely, x ∼ y if there exists an increasing homeomorphism h : R → R such that h ◦ f = f ◦ h and h(x) = y. An element x ∈ R is ordinary in (R,f) if its equivalence class [x] is a neighbourhood of it. </div></div></div>

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Asymptotic equivalence in a dynamical system

Proceedings of the Japan Academy ◽

10.3792/pja/1195518518 ◽

1975 ◽

Vol 51 (7) ◽

pp. 540-544

Author(s):

Shigeo Kono

Keyword(s):

Dynamical System ◽

Asymptotic Equivalence

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Remarks on some fundamental results about higher-rank graphs and their C*-algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091512000338 ◽

2013 ◽

Vol 56 (2) ◽

pp. 575-597 ◽

Cited By ~ 13

Author(s):

Robert Hazlewood ◽

Iain Raeburn ◽

Aidan Sims ◽

Samuel B. G. Webster

Keyword(s):

Equivalence Relation ◽

Direct Proof ◽

Explicit Description ◽

C Algebras ◽

Infinite Path ◽

Higher Rank ◽

Commuting Squares

AbstractResults of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterization, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.

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