The structure of random homeomorphisms
- PDF / 414,612 Bytes
- 39 Pages / 429.408 x 636.768 pts Page_size
- 105 Downloads / 171 Views
THE STRUCTURE OF RANDOM HOMEOMORPHISMS BY AND
Udayan B. Darji
´rton Elekes Ma
Department of Mathematics University of Louisville Louisville, KY 40292, USA and Ashoka University Rajiv Gandhi Education City Kundli, Rai 131029, India email: [email protected] http://www.math.louisville.edu/˜darji
Alfr´ed R´enyi Institute of Mathematics Hungarian Academy of Sciences PO Box 127, 1364 Budapest, Hungary and Institute of Mathematics E¨ otv¨ os Lor´ and University 1117 Budapest, Hungary email: [email protected] www.renyi.hu/˜emarci
AND
AND
Kende Kalina
Viktor Kiss
Institute of Mathematics E¨ otv¨ os Lor´ and University 1117 Budapest, Hungary email: [email protected]
Alfr´ed R´enyi Institute of Mathematics Hungarian Academy of Sciences PO Box 127, 1364 Budapest, Hungary and Institute of Mathematics E¨ otv¨ os Lor´ and University 1117 Budapest, Hungary email: [email protected] AND
´n Vidnya ´nszky Zolta Kurt G¨ odel Research Center for Mathematical Logic, Universit¨ at Wien W¨ ahringer Strasse 25, 1090 Wien, Austria and Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences PO Box 127, 1364 Budapest e-mail: [email protected] www.logic.univie.ac.at/˜vidnyanszz77 The second, fourth and fifth authors were partially supported by the Na-
tional Research, Development and Innovation Office—NKFIH, grants no. 113047, no. 104178 and no. 124749. The fifth author was also partially supported by FWF Grant P29999. Received August 27, 2018 and in revised form May 13, 2019
75
76
U. B. DARJI ET AL.
Isr. J. Math.
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 (2 ) 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.
1. Introduction The study of generic elements of Polish groups is a flourishing field with a large number of applications; ssee the works of Kechris and Rosendal [20], Truss [26], and Glasner and Weiss [15] among others. It is natural to ask whether there exist measure theoretic analogues of these results. Unfortunately, on nonlocally compact groups there is no natural invariant σ-finite measure. However, a generalisation of the
Data Loading...
