Hostname: page-component-784d4fb959-r4kcs Total loading time: 0 Render date: 2025-07-14T12:18:11.228Z Has data issue: false hasContentIssue false
  • English
  • Français

A Transformation with Simple Spectrum which is not Rank One

Published online by Cambridge University Press:  20 November 2018

Andrés Del Junco
Andrés Del Junco*
Affiliation:
University of British Columbia, Vancouver, British Columbia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Following [10] an ergodic measure-preserving transformation is called rank one if it admits a sequence of approximating stacks. Rank one transformations have been studied in [1] and [2] where it was shown that any rank one transformation has simple spectrum. More generally it has been shown by Chacon [4] that a transformation of rank n has spectral multiplicity at most n. M. A. Akcoglu and J. R. Baxter have asked whether the converse is true. In particular: does simple spectrum imply rank one? In this paper we give a negative answer to this question.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 3 , 01 June 1977 , pp. 655 - 663
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Baxter, J. R., A class of ergodic automorphisms, Thesis, University of Toronto, 1969.Google Scholar
2. Baxter, J. R. A class of ergodic transformations having simple spectrum, Proc. Amer. Math. Soc. 24 (1970).Google Scholar
3. Chacon, R. V., Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc. 22 (1969), 559–562.Google Scholar
4. Chacon, R. V. Approximation and spectral multiplicity, Contributions to Ergodic Theory and Probability, Springer Lecture Notes 160 (1970), 18–27.Google Scholar
5. Friedman, N. A., Introduction to ergodic theory (Van Nostrand Reinhold, 1970).Google Scholar
6. Halmos, P. R., Ergodic theory (Chelsea, 1956).Google Scholar
7. del Junco, A., Transformations with discrete spectrum are stacking transformations, Can. J. Math. 28 (1976), 836–839.Google Scholar
8. Kakutani, S., On the ergodic theory of shift transformations, Proc. 5th Berkeley Symp. 2 part ii, 405–414. (University of Calif. Press, Berkeley, 1967).Google Scholar
9. Kakutani, S. Strictly ergodic symbolic dynamical systems, Proc. 6th Berkeley Symp. 2, 319–326 (University of Calif. Press, Berkeley, 1972).Google Scholar
10. Weiss, B., Equivalence of measure preserving transformations, preprint.Google Scholar