On the spectral multiplicity of a class of finite rank transformations

Goodson, G. R.

doi:10.1090/S0002-9939-1985-0770541-8

On the spectral multiplicity of a class of finite rank transformations
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by G. R. Goodson PDF

Proc. Amer. Math. Soc. 93 (1985), 303-306 Request permission

Abstract:

The rank $M$ transformations, which Chacon called the simple approximations with multiplicity $M$, were shown by Chacon to have maximal spectral multiplicity at most $M$, although no example was given where this bound is attained for $M > 1$. In this paper, for each natural number $M > 1$, we show how to construct a simple approximation with multiplicity $M$ which is ergodic and has maximal spectral multiplicity equal to $M - 1$.

References

J. R. Baxter, A class of ergodic transformations having simple spectrum, Proc. Amer. Math. Soc. 27 (1971), 275–279. MR 276440, DOI 10.1090/S0002-9939-1971-0276440-2
R. V. Chacon, Approximation and spectral multiplicity, Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970) Springer, Berlin, 1970, pp. 18–27. MR 0271303
Andrés del Junco, A transformation with simple spectrum which is not rank one, Canadian J. Math. 29 (1977), no. 3, 655–663. MR 466489, DOI 10.4153/CJM-1977-067-7
Nathaniel A. Friedman, Introduction to ergodic theory, Van Nostrand Reinhold Mathematical Studies, No. 29, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1970. MR 0435350
V. I. Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR 168 (1966), 1009–1011 (Russian). MR 0199347
William Parry, Topics in ergodic theory, Cambridge Tracts in Mathematics, vol. 75, Cambridge University Press, Cambridge-New York, 1981. MR 614142
E. A. Robinson Jr., Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), no. 2, 299–314. MR 700773, DOI 10.1007/BF01389325