scholarly journals Unique ergodicity of the automorphism group of the semigeneric directed graph

2021 ◽  
pp. 1-14
Author(s):  
COLIN JAHEL
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Unique Ergodicity ◽  
Uniquely Ergodic

Abstract We prove that the automorphism group of the semigeneric directed graph (in the sense of Cherlin’s classification) is uniquely ergodic.

scholarly journals Quantum Symmetries of Graph C*-algebras

2018 ◽  
Vol 61 (4) ◽  
pp. 848-864 ◽  
Author(s):  
Simon Schmidt ◽  
Moritz Weber
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Operator Algebras ◽  
Automorphism Groups ◽  
Undirected Graphs ◽  
C Algebras ◽  
Quantum Symmetry ◽  
Quantum Symmetries ◽  
Multiple Edges ◽  
Definition Of

AbstractThe study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

scholarly journals Finite Factors of Bernoulli Schemes and Distinguishing Labelings of Directed Graphs

10.37236/6 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Andrew Lazowski ◽  
Stephen M. Shea
Keyword(s):  
Automorphism Group ◽  
Lower Bound ◽  
Directed Graph ◽  
Directed Graphs ◽  
Bell Numbers ◽  
Undirected Graphs ◽  
Bernoulli Scheme ◽  
Finite Set ◽  
Finite Factor

A labeling of a graph is a function from the vertices of the graph to some finite set.  In 1996, Albertson and Collins defined distinguishing labelings of undirected graphs.  Their definition easily extends to directed graphs.  Let $G$ be a directed graph associated to the $k$-block presentation of a Bernoulli scheme $X$.  We determine the automorphism group of $G$, and thus the distinguishing labelings of $G$.  A labeling of $G$ defines a finite factor of $X$.  We define demarcating labelings and prove that demarcating labelings define finitarily Markovian finite factors of $X$.  We use the Bell numbers to find a lower bound for the number of finitarily Markovian finite factors of a Bernoulli scheme.  We show that demarcating labelings of $G$ are distinguishing.

On Certain K-Groups Associated with Minimal Flows

1998 ◽  
Vol 41 (2) ◽  
pp. 240-244
Author(s):  
Jingbo Xia
Keyword(s):  
Toeplitz Operators ◽  
General Setting ◽  
Unique Ergodicity ◽  
Toeplitz Algebra ◽  
Commutator Ideal ◽  
Uniquely Ergodic

AbstractIt is known that the Toeplitz algebra associated with any flow which is both minimal and uniquely ergodic always has a trivial K1-group. We show in this note that if the unique ergodicity is dropped, then such K1-group can be non-trivial. Therefore, in the general setting of minimal flows, even the K-theoretical index is not sufficient for the classification of Toeplitz operators which are invertible modulo the commutator ideal.

scholarly journals Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph

2016 ◽  
Vol 160 (3) ◽  
pp. 437-462 ◽  
Author(s):  
IGOR DOLINKA ◽  
ROBERT D. GRAY ◽  
JILLIAN D. McPHEE ◽  
JAMES D. MITCHELL ◽  
MARTYN QUICK
Keyword(s):  
Automorphism Group ◽  
Bipartite Graph ◽  
Random Graph ◽  
Directed Graph ◽  
Automorphism Groups ◽  
Structural Information ◽  
Maximal Subgroups ◽  
Endomorphism Monoid

AbstractWe establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph R. As a consequence we show that, for any countable graph Γ, there are uncountably many maximal subgroups of the endomorphism monoid of R isomorphic to the automorphism group of Γ. Further structural information about End R is established including that Aut Γ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.

scholarly journals Amenability and unique ergodicity of automorphism groups of countable homogeneous directed graphs

2018 ◽  
Vol 40 (5) ◽  
pp. 1351-1401
Author(s):  
MICHEAL PAWLIUK ◽  
MIODRAG SOKIĆ
Keyword(s):  
Automorphism Groups ◽  
Directed Graphs ◽  
Point Of View ◽  
Unique Ergodicity ◽  
Minimal Flow ◽  
Negative Results ◽  
Starting Point ◽  
Complete Multipartite ◽  
Uniquely Ergodic

We study the automorphism groups of countable homogeneous directed graphs (and some additional homogeneous structures) from the point of view of topological dynamics. We determine precisely which of these automorphism groups are amenable (in their natural topologies). For those which are amenable, we determine whether they are uniquely ergodic, leaving unsettled precisely one case (the ‘semi-generic’ complete multipartite directed graph). We also consider the Hrushovski property. For most of our results we use the various techniques of Angelet al[Random orderings and unique ergodicity of automorphism groups.J. Eur. Math. Soc.,16(2014), 2059–2095], suitably generalized to a context in which the universal minimal flow is not necessarily the space of all orders. Negative results concerning amenability rely on constructions of the type considered in Zucker [Amenability and unique ergodicity of automorphism groups of Fraïssé structures.Fund. Math.,226(2014), 41–61]. An additional class of structures (compositions) may be handled directly on the basis of very general principles. The starting point in all cases is the determination of the universal minimal flow for the automorphism group, which in the context of countable homogeneous directed graphs is given in Jasińskiet al[Ramsey precompact expansions of homogeneous directed graphs.Electron. J. Combin.,21(4), (2014), 31] and the papers cited therein.

scholarly journals AUTOMORPHISMS OF NONSELFADJOINT DIRECTED GRAPH OPERATOR ALGEBRAS

2009 ◽  
Vol 87 (2) ◽  
pp. 175-196
Author(s):  
BENTON L. DUNCAN
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Maximal Ideal ◽  
Operator Algebras ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Normal Subgroups ◽  
Inner Automorphisms

AbstractWe analyze the automorphism group for the norm closed quiver algebras 𝒯+(Q). We begin by focusing on two normal subgroups of the automorphism group which are characterized by their actions on the maximal ideal space of 𝒯+(Q). To further discuss arbitrary automorphisms we factor automorphism through subalgebras for which the automorphism group can be better understood. This allows us to classify a large number of noninner automorphisms. We suggest a candidate for the group of inner automorphisms.

SEMILATTICES AND THE RAMSEY PROPERTY

2015 ◽  
Vol 80 (4) ◽  
pp. 1236-1259 ◽  
Author(s):  
MIODRAG SOKIĆ
Keyword(s):  
Automorphism Group ◽  
Minimal Flow ◽  
Linear Extensions ◽  
Ramsey Property ◽  
Linear Orderings ◽  
Topological Interpretation ◽  
Finite Lattices ◽  
Image Position ◽  
Fraïssé Class ◽  
Uniquely Ergodic

AbstractWe consider${\cal S}$, the class of finite semilattices;${\cal T}$, the class of finite treeable semilattices; and${{\cal T}_m}$, the subclass of${\cal T}$which contains trees with branching bounded bym. We prove that${\cal E}{\cal S}$, the class of finite lattices with linear extensions, is a Ramsey class. We calculate Ramsey degrees for structures in${\cal S}$,${\cal T}$, and${{\cal T}_m}$. In addition to this we give a topological interpretation of our results and we apply our result to canonization of linear orderings on finite semilattices. In particular, we give an example of a Fraïssé class${\cal K}$which is not a Hrushovski class, and for which the automorphism group of the Fraïssé limit of${\cal K}$is not extremely amenable (with the infinite universal minimal flow) but is uniquely ergodic.

scholarly journals Unique ergodicity for non-uniquely ergodic horocycle flows

2006 ◽  
Vol 16 (2) ◽  
pp. 411-433 ◽  
Author(s):  
François Ledrappier ◽  
◽  
Omri Sarig ◽  
Keyword(s):  
Unique Ergodicity ◽  
Uniquely Ergodic

scholarly journals Boshernitzan's criterion for unique ergodicity of an interval exchange transformation

1987 ◽  
Vol 7 (1) ◽  
pp. 149-153 ◽  
Author(s):  
William A. Veech
Keyword(s):  
Unique Ergodicity ◽  
Interval Exchange ◽  
Interval Exchange Transformation ◽  
Exchange Transformation ◽  
Uniquely Ergodic

AbstractConfirming a conjecture by Boshernitzan, it is proved that ifTis a minimal non-uniquely ergodic interval exchange, the minimum spacing of the partition determined byTnis O(1/n).

A context in which finite or unique ergodicity is generic

2020 ◽  
pp. 1-23
Author(s):  
ANDY Q. YINGST
Keyword(s):  
Unique Ergodicity ◽  
Cantor Space ◽  
Bernoulli Trial ◽  
Residual Set ◽  
Ergodic Measures ◽  
Uniquely Ergodic

Abstract We show that for good measures, the set of homeomorphisms of Cantor space which preserve that measure and which have no invariant clopen sets contains a residual set of homeomorphisms which are uniquely ergodic. Additionally, we show that for refinable Bernoulli trial measures, the same set of homeomorphisms contains a residual set of homeomorphisms which admit only finitely many ergodic measures.

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