The prevalence of permutations with infinite cycles

Randall Dougherty; Jan Mycielski

Fundamenta Mathematicae (1994)

  • Volume: 144, Issue: 1, page 89-94
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Dougherty, Randall, and Mycielski, Jan. "The prevalence of permutations with infinite cycles." Fundamenta Mathematicae 144.1 (1994): 89-94. <http://eudml.org/doc/212017>.

@article{Dougherty1994,
abstract = {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.},
author = {Dougherty, Randall, Mycielski, Jan},
journal = {Fundamenta Mathematicae},
keywords = {Polish group; prevalent; Borel probability measure; Haar measure; compact group; shy set; permutations; cycles},
language = {eng},
number = {1},
pages = {89-94},
title = {The prevalence of permutations with infinite cycles},
url = {http://eudml.org/doc/212017},
volume = {144},
year = {1994},
}

TY - JOUR
AU - Dougherty, Randall
AU - Mycielski, Jan
TI - The prevalence of permutations with infinite cycles
JO - Fundamenta Mathematicae
PY - 1994
VL - 144
IS - 1
SP - 89
EP - 94
AB - 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.
LA - eng
KW - Polish group; prevalent; Borel probability measure; Haar measure; compact group; shy set; permutations; cycles
UR - http://eudml.org/doc/212017
ER -

References

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  1. [1] J. P. R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255-260. 
  2. [2] R. Dougherty, Examples of non-shy sets, this volume, 73-88. Zbl0842.43006
  3. [3] B. R. Hunt, The prevalence of continuous nowhere differentiable functions, to appear. Zbl0861.26003
  4. [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. Zbl0763.28009
  5. [5] D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience Publ., New York, 1955. Zbl0068.01904
  6. [6] J. Mycielski, Some unsolved problems on the prevalence of ergodicity, instability and algebraic independence, Ulam Quart. 1 (3) (1992), 30-37. Zbl0846.28006

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