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

Part of: Abstract harmonic analysis Connections with other structures, applications Graph theory

Published online by Cambridge University Press:  17 October 2018

MICHEAL PAWLIUK  and
MIODRAG SOKIĆ
MICHEAL PAWLIUK
Affiliation:
University of CalgaryMathematics & Statistics, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 email mpawliuk@ucalgary.ca
MIODRAG SOKIĆ
Affiliation:
University of CalgaryMathematics & Statistics, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 email mpawliuk@ucalgary.ca
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Abstract

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 Angel et 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 Jasiński et al [Ramsey precompact expansions of homogeneous directed graphs. Electron. J. Combin., 21(4), (2014), 31] and the papers cited therein.

Keywords

infinite dimensional dynamicstopological dynamicsamenabilitydirected graphs (digraphs)tournaments

MSC classification

Primary: 43A07: Means on groups, semigroups, etc.; amenable groups
Secondary: 05C55: Generalized Ramsey theory 54H20: Topological dynamics 05C20: Directed graphs (digraphs), tournaments
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 5 , May 2020 , pp. 1351 - 1401
Copyright
© Cambridge University Press, 2018

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