Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-07-04T20:12:52.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixing and spectral multiplicity

Published online by Cambridge University Press:  19 September 2008

E. Arthur Robinson Jr
E. Arthur Robinson Jr
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA
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.

A mixing transformation is constructed which has non-simple spectrum of finite multiplicity. This example is based on a rank 1 mixing transformation and is constructed by cutting and stacking. It can be made to be mixing of all orders.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 4 , December 1985 , pp. 617 - 624
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Baxter, J. R.. A class of ergodic transformations having simple spectrum. Proc. Amer. Math. Soc. 27 (1971), 275–279.CrossRefGoogle Scholar
[2]Berg, K. R.. Mixing, cyclic approximation, and homomorphisms. Preprint, University of Maryland.Google Scholar
[3]Chacon, R. V., Approximation of transformations with continuous spectrum. Pacific J. Math. 2 (1969), 293–302.CrossRefGoogle Scholar
[4]Chacon, R. V.. Approximation and spectral multiplicity. Springer Lect. Notes in Math. 160 (1970), 18–22.CrossRefGoogle Scholar
[5]Friedman, N. A.. Introduction to Ergodic Theory. Van Nostrand Reinhold Co.: New York (1970).Google Scholar
[6]Goodson, G. R.. On the spectral multiplicity of a class of finite rank transformations. Preprint, University of Capetown.Google Scholar
[7]Jones, R. & Parry, W.. Compact abelian group extensions of dynamical systems, II. Comp. Math. 25 (1972), 135–147.Google Scholar
[8]del Junco, A.. A transformation with simple spectrum which is not rank one. Can J. Math. 29 (1977), 655–663.CrossRefGoogle Scholar
[9]Kalikow, S.. Two fold mixing implies three fold mixing for rank one transformations. Ergod. Th. & Dynam. Sys. 4 (1984), 237–259.CrossRefGoogle Scholar
[10]Katok, A.. Constructions in ergodic theory. To appear in Progress in Mathematics, Birkhauser, Boston, Mass.Google Scholar
[11]Mathew, J. & Nadkarni, M. G.. A measure preserving transformation whose spectrum has Lebesgue component of multiplicity two. Bull London Math. Soc. 16 (1984), 402–406CrossRefGoogle Scholar
[12]Ornstein, D. S.. On the root problem in ergodic theory. In Proceedings of the Sixth Berkeley Symposium on Statistics and Probability, Vol. II, University of California Press (1967), 1–49.Google Scholar
[13]Oseledec, V. I.. The spectrum of ergodic automorphisms. Dokl Akad. Nauk SSSR 168 (1966), 776–779.Google Scholar
[14]Parry, W.. Ergodic properties of affine transformations and flows on nilmanifolds. Amer. J. Math. 91 (1961), 757–771.CrossRefGoogle Scholar
[15]Robinson, E. A.. Ergodic measure preserving transformations with arbitrary finite spectral multiplicities. Inv. Math. 72 (1983), 299–314.CrossRefGoogle Scholar
[16]Robinson, E. A.. Ergodic measure preserving transformations with finite spectral multiplicities. Dissertation, University of Maryland, 08 1983.CrossRefGoogle Scholar
[17]Robinson, E. A.. Transformations with highly non-homogeneous spectrum of finite multiplicity. In preparation.Google Scholar
[18]Rudolph, D. J.. Classifying the isometric extensions of a Bernoulli shift. J. D'Analyse Math. 34 (1978), 36–60.CrossRefGoogle Scholar
[19]Rudolph, D. J.. An example of a measure preserving transformation with minimal self-joinings and applications. J. D'Analyse Math. 35 (1979), 97–122.CrossRefGoogle Scholar
[20]Rudolph, D. J.. K-fold mixing lifts to weakly mixing isometric extensions. Preprint, University of Maryland.Google Scholar