# The generic transformation has roots of all orders

Colloquium Mathematicae (2000)

• Volume: 84/85, Issue: 2, page 521-547
• ISSN: 0010-1354

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## Abstract

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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.

## How to cite

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King, Jonathan. "The generic transformation has roots of all orders." Colloquium Mathematicae 84/85.2 (2000): 521-547. <http://eudml.org/doc/210831>.

@article{King2000,
abstract = {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.},
author = {King, Jonathan},
journal = {Colloquium Mathematicae},
keywords = {generic measure-preserving transformation; roots of all orders},
language = {eng},
number = {2},
pages = {521-547},
title = {The generic transformation has roots of all orders},
url = {http://eudml.org/doc/210831},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - King, Jonathan
TI - The generic transformation has roots of all orders
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 2
SP - 521
EP - 547
AB - 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.
LA - eng
KW - generic measure-preserving transformation; roots of all orders
UR - http://eudml.org/doc/210831
ER -

## References

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12. [12] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966.
13. [13] B. F. Madore, Rank-one mixing actions with simple ℤ subactions, Doctoral dissertation, 2000.
14. [14] D. S. Ornstein, On the root problem in ergodic theory, in: Proc. Sixth Berkeley Sympos. Math. Statist. Probab. (Berkeley, CA, 1970/1971), Vol. II: Probability Theory, Univ. California Press, Berkeley, CA, 1972, 347-356.
15. [15] D. J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Anal. Math. 35:97-122, 1979 Zbl0446.28018
16. [16] W. A. Veech, A criterion for a process to be prime, Monatsh. Math. 94:335-341, 1982. Zbl0499.28016

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