In the sense of the Baire Category Theorem we show that the generic transformation T has roots of all orders (RAO theorem). The argument appears novel in that it proceeds by establishing that the set of such T is not meager - and then appeals to a Zero-One Law (Lemma 2). On the group Ω of (invertible measure-preserving) transformations, §D shows that the squaring map p: S → S^{2} is topologically complex in that both the locally-dense and locally-lacunary points of p are dense (Theorem 23). The last section, §E, discusses the relation between RAO and a recent example of Blair Madore. Answering a question of the author's, Madore constructs a transformation with a square-root chain of each finite length, yet possessing no infinite square-root chain.
@article{bwmeta1.element.bwnjournal-article-cmv84i2p521bwm,
author = {Jonathan King},
title = {The generic transformation has roots of all orders},
journal = {Colloquium Mathematicae},
volume = {84/85},
year = {2000},
pages = {521-547},
zbl = {0972.37001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv84i2p521bwm}
}
King, Jonathan. The generic transformation has roots of all orders. Colloquium Mathematicae, Tome 84/85 (2000) pp. 521-547. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv84i2p521bwm/
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