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On the existence of automorphisms with simple Lebesgue spectrum

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Abstract

It is shown that ifT is a measure preserving automorphism on a probability space (Ω,B, m) which admits a random variable X0 with mean zero such that the stochastic sequence X0 o Tn,n ε ℤ is orthonormal and spans L0 2(Ω,B,m), then for any integerk ≠ 0, the random variablesX o Tnk,n ε ℤ generateB modulom.

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References

  1. D S Ornstein, On the root problem in ergodic theory,Proc. Sixth Berkeley Symp. Math. Stat. and Probability, Vol. H, pp. 347–356, University of California Press, 1972

  2. J B Robertson, A two state pairwise independent stationary process for which X1,X3,X5 are dependent,Sankhya, Indian J. Stat., Ser. A, t. 50 (1988) 171–183

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Authors and Affiliations

  1. Institut Galilée, Université Paris XIII, 93430, Villetaneuse cedex, France

    A. Kłopotowski & M. G. Nadkarni

  2. Department of Mathematics, University of Mumbai, 400098, Kalina, Mumbai, India

    M. G. Nadkarni

Authors
  1. A. Kłopotowski
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  2. M. G. Nadkarni
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Correspondence to A. Kłopotowski.

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Kłopotowski, A., Nadkarni, M.G. On the existence of automorphisms with simple Lebesgue spectrum. Proc. Indian Acad. Sci. (Math. Sci.) 109, 47–55 (1999). https://doi.org/10.1007/BF02837766

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  • DOI: https://doi.org/10.1007/BF02837766

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