2017/12/26 by Micheal Pawliuk, Pawliuk, Micheal, Miodrag Sokić +1 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1712.09461
openalex publication_date 2017/12/26 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We study the automorphism groups of countable homogeneous directed graphs\n(and some additional homogeneous structures) from the point of view of\ntopological dynamics. We determine precisely which of these automorphism groups\nare amenable (in their natural topologies). For those which are amenable, we\ndetermine whether they are uniquely ergodic, leaving unsettled precisely one\ncase (the "semi-generic" complete multipartite directed graph). We also\nconsider the Hrushovski property. For most of our results we use the various\ntechniques of [3], suitably generalized to a context in which the universal\nminimal flow is not necessarily the space of all orders. Negative results\nconcerning amenability rely on constructions of the type considered in [26]. An\nadditional class of structures (compositions) may be handled directly on the\nbasis of very general principles. The starting point in all cases is the\ndetermination of the universal minimal flow for the automorphism group, which\nin the context of countable homogeneous directed graphs is given in [10] and\nthe papers cited therein.\n