A number of recent papers have been devoted to the study of prevalence, a generalization of the property of being of full Haar measure to topological groups which need not have a Haar measure, and the dual concept of shyness. These concepts give a notion of "largeness" which often differs from the category analogue, comeagerness, and may be closer to the intuitive notion of "almost everywhere." In this paper, we consider the group of permutations of natural numbers. Here, in the sense of category, "almost all" permutations have only finite cycles. In contrast, we show that, in terms of prevalence, "almost all" permutations have infinitely many infinite cycles and only finitely many finite cycles; this set of permutations comprises countably many conjugacy classes, each of which is non-shy.
@article{bwmeta1.element.bwnjournal-article-fmv144i1p89bwm,
author = {Randall Dougherty and Jan Mycielski},
title = {The prevalence of permutations with infinite cycles},
journal = {Fundamenta Mathematicae},
volume = {144},
year = {1994},
pages = {89-94},
zbl = {0837.43008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv144i1p89bwm}
}
Dougherty, Randall; Mycielski, Jan. The prevalence of permutations with infinite cycles. Fundamenta Mathematicae, Tome 144 (1994) pp. 89-94. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv144i1p89bwm/
[00000] [1] J. P. R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255-260.
[00001] [2] R. Dougherty, Examples of non-shy sets, this volume, 73-88. | Zbl 0842.43006
[00002] [3] B. R. Hunt, The prevalence of continuous nowhere differentiable functions, to appear. | Zbl 0861.26003
[00003] [4] B. R. Hunt, T. Sauer, and J. A. Yorke, Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces, Bull. Amer. Math. Soc. 27 (1992), 217-238. | Zbl 0763.28009
[00004] [5] D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience Publ., New York, 1955. | Zbl 0068.01904
[00005] [6] J. Mycielski, Some unsolved problems on the prevalence of ergodicity, instability and algebraic independence, Ulam Quart. 1 (3) (1992), 30-37. | Zbl 0846.28006