×

Random orderings and unique ergodicity of automorphism groups. (English) Zbl 1304.22027

Ergodic theory is studied of the potential phenomenon. Let \(\Gamma\) be a topological group. A \(\Gamma\)-flow is uniquely ergodic if it has a unique ergodic measure. Every topological group \(\Gamma\) admits a unique, up to isomorphism, minimal flow \(M(\Gamma)\). When \(\Gamma\) is compact, \(M(\Gamma)=\Gamma\). It is shown that for certain automorphism groups every minimal action, but also this measure, concentrates on a single comeager orbit. Finally some open problems are discussed.

MSC:

22F50 Groups as automorphisms of other structures
03C98 Applications of model theory
05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
28D15 General groups of measure-preserving transformations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Becker, H., Kechris, A. S.: The Descriptive Set Theory of Polish Group Actions. Lon- don Math. Soc. Lecture Note Ser. 232. Cambridge Univ. Press, Cambridge (1996) · Zbl 0949.54052
[2] Bekka, B., de la Harpe, P., Valette, A.: Kazhdan’s Property (T). New Math. Monogr. 11, Cambridge Univ. Press, Cambridge (2008) · Zbl 1146.22009
[3] Fraïssé, R.: Sur l’extension aux relations de quelques propriétés des ordres. Ann. Sci. École Norm. Sup. (3) 71, 363-388 (1954) · Zbl 0057.04206
[4] Glasner, E.: Ergodic Theory via Joinings. Amer. Math. Soc. (2003) · Zbl 1038.37002
[5] Glasner, E., Weiss, B.: Minimal actions of the group S(Z) of permutations of the integers, Geom. Funct. Anal. 12, 964-988 (2002) · Zbl 1025.37006 · doi:10.1007/PL00012651
[6] Glasner, E., Weiss, B.: The universal minimal system for the group of homeomorphisms of the Cantor set. Fund. Math. 176, 277-289 (2003) · Zbl 1022.37009 · doi:10.4064/fm176-3-6
[7] Herwig, B.: Extending partial isomorphisms on finite structures. Combinatorica 15, 365- 371 (1995) · Zbl 0830.05037 · doi:10.1007/BF01299742
[8] Herwig, B.: Extending partial isomorphisms for the small index property of many \omega - categorical structures. Israel J. Math. 107, 93-123 (1998) · Zbl 0922.03044 · doi:10.1007/BF02764005
[9] Hodges, W.: Model Theory. Cambridge Univ. Press (1993) · Zbl 0789.03031
[10] Hrushovski, E.: Extending partial isomorphisms of graphs. Combinatorica 12, 411-416 (1992) · Zbl 0767.05053 · doi:10.1007/BF01305233
[11] Jasiński, J.: Hrushovski and Ramsey properties of classes of finite inner product structures, finite Euclidean metric spaces, and boron trees, Ph.D. Thesis, Univ. of Toronto (2011)
[12] Kechris, A. S.: Classical Descriptive Set Theory. Springer (1995) · Zbl 0819.04002
[13] Kechris, A. S., Pestov, V. G., Todorcevic, S.: Fraïssé limits, Ramsey theory, and topo- logical dynamics of automorphism groups. Geom. Funct. Anal. 15, 106-189 (2005) · Zbl 1084.54014 · doi:10.1007/s00039-005-0503-1
[14] Kechris, A. S., Rosendal, C.: Turbulence, amalgamation, and generic automorphisms of homogeneous structures. Proc. London Math. Soc. 94, 302-350 (2007) · Zbl 1118.03042 · doi:10.1112/plms/pdl007
[15] Kechris, A. S., Sokić, M.: Dynamical properties of the automorphism groups of the random poset and random distributive lattice. Fund. Math. 218, 69-94 (2012) · Zbl 1260.03063 · doi:10.4064/fm218-1-4
[16] McDiarmid, C.: On the method of bounded differences. In: J. Siemons (ed.), Surveys in Combinatorics, 1989, London Math. Soc. Lecture Note Ser. 141, Cambridge Univ. Press, 148-188 (1989) · Zbl 0712.05012
[17] Meckes, M. W.: Positive definite metric spaces. Positivity 17, 733-757 (2013) · Zbl 1284.54043 · doi:10.1007/s11117-012-0202-8
[18] Melleray, J., Tsankov, T.: Generic representations of abelian groups and extreme amenabil- ity. Israel J. Math. 198, 129-167 (2013) · Zbl 1279.43002 · doi:10.1007/s11856-013-0036-5
[19] Moore, J. T.: Amenability and Ramsey theory. Fund. Math. 220, 263-280 (2013) · Zbl 1263.05114 · doi:10.4064/fm220-3-6
[20] Ne\check set\check ril, J., Rödl, V.: On a probabilistic graph-theoretical method. Proc. Amer. Math. Soc. 72, 417-421 (1978) · Zbl 0399.05007 · doi:10.2307/2042818
[21] Nguyen Van Thé, L.: Structural Ramsey theory of metric spaces and topological dynam- ics of isometry groups. Mem. Amer. Math. Soc. 206, no. 968 (2010) · Zbl 1203.05159 · doi:10.1090/S0065-9266-10-00586-7
[22] Nguyen Van Thé, L.: More on the Kechris-Pestov-Todorcevic correspondence: Precom- pact expansions. Fund. Math. 222, 19-47 (2013) · Zbl 1293.37006 · doi:10.4064/fm222-1-2
[23] Pestov, V.: On free actions, minimal flows and a problem by Ellis. Trans. Amer. Math. Soc. 350, 4149-4165 (1998) · Zbl 0911.54034 · doi:10.1090/S0002-9947-98-02329-0
[24] Pestov, V.: Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups. Israel J. Math. 127, 317-357 (2002) · Zbl 1007.43001 · doi:10.1007/BF02784537
[25] Schoenberg, I. J.: Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44, 522-536 (1938) · Zbl 0019.41502 · doi:10.2307/1989894
[26] Solecki, S.: Extending partial isometries. Israel J. Math. 150, 315-332 (2005) · Zbl 1124.54012 · doi:10.1007/BF02762385
[27] Weiss, B.: Minimal models for free actions. In: Dynamical Systems and Group Actions, Contemp. Math. 567, Amer. Math. Soc., 249-264 (2012) · Zbl 1279.37010 · doi:10.1090/conm/567/11253
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.