# 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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topKing, 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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- [11] J. L. King, Joining-rank and the structure of finite rank mixing transformations, J. Anal. Math. 51:182-227, 1988. Zbl0665.28010
- [12] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966.
- [13] B. F. Madore, Rank-one mixing actions with simple ℤ subactions, Doctoral dissertation, 2000.
- [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] 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] W. A. Veech, A criterion for a process to be prime, Monatsh. Math. 94:335-341, 1982. Zbl0499.28016

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