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The generic transformation has roots of all orders

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King J. L. F.

Colloquium Mathematicum

2000

tom 84/85

nr 2

521-547

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.

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1 Colloquium Mathematicum

1 King J. L. F.

1 2000

Wyniki wyszukiwania

The generic transformation has roots of all orders