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

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