The universal minimal flow of the homeomorphism group of the Lelek fan
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- by Dana Bartošová and Aleksandra Kwiatkowska PDF
- Trans. Amer. Math. Soc. 371 (2019), 6995-7027 Request permission
Abstract:
We compute the universal minimal flow of the homeomorphism group of the Lelek fan—a one-dimensional tree-like continuum with many symmetries.References
- Dana Bartošová and Aleksandra Kwiatkowska, Gowers’ Ramsey theorem with multiple operations and dynamics of the homeomorphism group of the Lelek fan, J. Combin. Theory Ser. A 150 (2017), 108–136. MR 3645569, DOI 10.1016/j.jcta.2017.02.002
- Dana Bartošová and Aleksandra Kwiatkowska, Lelek fan from a projective Fraïssé limit, Fund. Math. 231 (2015), no. 1, 57–79. MR 3361235, DOI 10.4064/fm231-1-4
- D. Bartošová, J. Lopez-Abad, M. Lupini, and B. Mbombo, Ramsey properties for finite dimensional normed spaces, arXiv:1708.01317 (2017).
- Howard Becker and Alexander S. Kechris, The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996. MR 1425877, DOI 10.1017/CBO9780511735264
- Witold D. Bula and Lex G. Oversteegen, A characterization of smooth Cantor bouquets, Proc. Amer. Math. Soc. 108 (1990), no. 2, 529–534. MR 991691, DOI 10.1090/S0002-9939-1990-0991691-9
- Włodzimierz J. Charatonik, The Lelek fan is unique, Houston J. Math. 15 (1989), no. 1, 27–34. MR 1002079
- E. Glasner and B. Weiss, The universal minimal system for the group of homeomorphisms of the Cantor set, Fund. Math. 176 (2003), no. 3, 277–289. MR 1992824, DOI 10.4064/fm176-3-6
- E. Glasner and B. Weiss, Minimal actions of the group $\Bbb S(\Bbb Z)$ of permutations of the integers, Geom. Funct. Anal. 12 (2002), no. 5, 964–988. MR 1937832, DOI 10.1007/PL00012651
- R. L. Graham and B. L. Rothschild, Ramsey’s theorem for $n$-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257–292. MR 284352, DOI 10.1090/S0002-9947-1971-0284352-8
- Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
- Wilfrid Hodges, A shorter model theory, Cambridge University Press, Cambridge, 1997. MR 1462612
- Trevor Irwin and Sławomir Solecki, Projective Fraïssé limits and the pseudo-arc, Trans. Amer. Math. Soc. 358 (2006), no. 7, 3077–3096. MR 2216259, DOI 10.1090/S0002-9947-06-03928-6
- A. S. Kechris, V. G. Pestov, and S. Todorcevic, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal. 15 (2005), no. 1, 106–189. MR 2140630, DOI 10.1007/s00039-005-0503-1
- A. Lelek, On plane dendroids and their end points in the classical sense, Fund. Math. 49 (1960/61), 301–319. MR 133806, DOI 10.4064/fm-49-3-301-319
- L. Nguyen Van Thé, More on the Kechris-Pestov-Todorcevic correspondence: precompact expansions, Fund. Math. 222 (2013), no. 1, 19–47. MR 3080786, DOI 10.4064/fm222-1-2
- Vladimir Pestov, Dynamics of infinite-dimensional groups, University Lecture Series, vol. 40, American Mathematical Society, Providence, RI, 2006. The Ramsey-Dvoretzky-Milman phenomenon; Revised edition of Dynamics of infinite-dimensional groups and Ramsey-type phenomena [Inst. Mat. Pura. Apl. (IMPA), Rio de Janeiro, 2005; MR2164572]. MR 2277969, DOI 10.1090/ulect/040
- Vladimir G. Pestov, On free actions, minimal flows, and a problem by Ellis, Trans. Amer. Math. Soc. 350 (1998), no. 10, 4149–4165. MR 1608494, DOI 10.1090/S0002-9947-98-02329-0
- Vladimir Uspenskij, On universal minimal compact $G$-spaces, Proceedings of the 2000 Topology and Dynamics Conference (San Antonio, TX), 2000, pp. 301–308. MR 1875600
- Andy Zucker, Topological dynamics of automorphism groups, ultrafilter combinatorics, and the generic point problem, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6715–6740. MR 3461049, DOI 10.1090/tran6685
Additional Information
- Dana Bartošová
- Affiliation: Institute de Matematica e Estatística, Universidade de São Paulo, Brazil; and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania
- Email: dbartoso@andrew.cmu.edu
- Aleksandra Kwiatkowska
- Affiliation: Institut für Mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany; and Instytut Matematyczny, Uniwersytet Wrocławski, plac Grunwaldzki 2/4, 50-384 Wrocław, Poland
- Email: kwiatkoa@uni-muenster.de
- Received by editor(s): June 28, 2017
- Received by editor(s) in revised form: February 6, 2018
- Published electronically: September 25, 2018
- Additional Notes: The first author was supported by FAPESP (grant 2013/14458-9) and FAPESP (grant 2014/12405-8).
The second author was supported by Narodowe Centrum Nauki grant 2016/23/D/ST1/01097. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 6995-7027
- MSC (2010): Primary 05D10, 37B05, 54F15, 03C98
- DOI: https://doi.org/10.1090/tran/7548
- MathSciNet review: 3939568