The generic transformation has roots of all orders
Colloquium Mathematicum, Tome 84 (2000) no. 2, pp. 521-547
Cet article a éte moissonné depuis 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.
@article{10_4064_cm_84_85_2_521_547,
author = {Jonathan King},
title = {The generic transformation has roots of all orders},
journal = {Colloquium Mathematicum},
pages = {521--547},
year = {2000},
volume = {84},
number = {2},
doi = {10.4064/cm-84/85-2-521-547},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm-84/85-2-521-547/}
}
TY - JOUR AU - Jonathan King TI - The generic transformation has roots of all orders JO - Colloquium Mathematicum PY - 2000 SP - 521 EP - 547 VL - 84 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4064/cm-84/85-2-521-547/ DO - 10.4064/cm-84/85-2-521-547 LA - en ID - 10_4064_cm_84_85_2_521_547 ER -
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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