Abstract
In order to understand the structure of the “typical” element of a homeomorphism group, one has to study how large the conjugacy classes of the group are. When typical means generic in the sense of Baire category, this is well understood; see, e.g., the works of Glasner and Weiss, and Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behaviour of the homeomorphisms is entirely different from the generic case. For Homeo+([0, 1]) we describe the non-Haar null conjugacy classes and also show that their union is co-Haar null, for Homeo+(S1) we describe the non-Haar null conjugacy classes, and for U(l2) we show that, apart from the classes of the multishifts, all conjugacy classes are Haar null. As an application we affirmatively answer the question whether these groups can be written as the union of a meagre and a Haar null set.
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Acknowledgements
We would like to thank R. Balka, Z. Gyenis, A. Kechris, C. Rosendal, S. Solecki and P. Wesolek for many valuable remarks and discussions.
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The second, fourth and fifth authors were partially supported by the National Research, Development and Innovation Office—NKFIH, grants no. 113047, no. f 04178 and no. 124749. The fifth author was also partially supported by FWF Grant P29999.
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Darji, U.B., Elekes, M., Kalina, K. et al. The structure of random homeomorphisms. Isr. J. Math. 237, 75–113 (2020). https://doi.org/10.1007/s11856-020-2001-4
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DOI: https://doi.org/10.1007/s11856-020-2001-4