Geodesic
The generic transformation has roots of all orders
Colloquium Mathematicum, Tome 84 (2000) no. 2, pp. 521-547.

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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.
DOI : 10.4064/cm-84/85-2-521-547

Jonathan King 1

1 
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Jonathan King. The generic transformation has roots of all orders. Colloquium Mathematicum, Tome 84 (2000) no. 2, pp. 521-547. doi : 10.4064/cm-84/85-2-521-547. http://geodesic.mathdoc.fr/articles/10.4064/cm-84/85-2-521-547/

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