The generic transformation has roots of all orders

Jonathan King

Colloquium Mathematicae (2000)

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

Abstract

top
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

top

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

top
  1. [1] F. Beleznay, The complexity of the collection of countable linear orders of the form (I+I), preprint, 1997. Zbl0944.03046
  2. [2] K. R. Berg, Mixing, cyclic approximation, and homomorphisms, Maryland Technical Report, 1975. 
  3. [3] N. Friedman, P. Gabriel, and J. L. King, An invariant for rigid rank-1 transformations, Ergodic Theory Dynam. Systems 8:53-72, 1988. Zbl0621.28011
  4. [4] E. Glasner and J. L. King, A zero-one law for dynamical properties, in: Topological Dynamics and Applications (Minneapolis, MN, 1995), Amer. Math. Soc., Providence, RI, 1998, 231-242. Zbl0909.28014
  5. [5] P. R. Halmos, Lectures on Ergodic Theory, Publ. Math. Soc. Japan 3, Math. Soc. Japan, 1956. 
  6. [6] P. D. Humke and M. Laczkovich, The Borel structure of iterates of continuous functions, Proc. Edinburgh Math. Soc. (2) 32:483-494, 1989. Zbl0671.28003
  7. [7] A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems 7:531-557, 1987. Zbl0646.60010
  8. [8] A. B. Katok, Ya. G. Sinaĭ, and A. M. Stepin, The theory of dynamical systems and general transformation groups with invariant measure, (errata insert), Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Akad. Nauk SSSR, Moscow, 1975, 129-262 (in Russian). Zbl0399.28011
  9. [9] A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspekhi Mat. Nauk 22 (1967), no. 5, 81-106. Zbl0172.07202
  10. [10] J. L. King, The commutant is the weak closure of the powers, for rank-1 transformations, Ergodic Theory Dynam. Systems 6:363-384, 1986. Zbl0595.47005
  11. [11] J. L. King, Joining-rank and the structure of finite rank mixing transformations, J. Anal. Math. 51:182-227, 1988. Zbl0665.28010
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.