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Subsequences of normal sequences

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Abstract

In this paper, we characterize a set of indices τ={τ(0)<τ(1)<…} such that forany normal sequence (α(0), α(1),…) of a certain type, the subsequence (α(τ(0)), α(τ(1)),…) is a normal sequence of the same type. Assume thatn→∞. Then, we prove that τ has this property if and only if the 0–1 sequence (θ τ (0), whereθ τ (i)=1 or 0 according asi∈{τ(j);j=0, 1,…} or not, iscompletely deterministic in the sense of B. Weiss.

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References

  1. L. M. Abramov,The entropy of the derived automorphism, Dokl. Akad. Nauk. SSSR128 (1959), 647–650.

    MATH  MathSciNet  Google Scholar 

  2. P. Billingsley,Ergodic Theory and Information, John Wiley and Sons, New York, 1965.

    MATH  Google Scholar 

  3. H. Furstenberg,Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1–1 (1967), 1–49.

    Article  MathSciNet  Google Scholar 

  4. W. H. Gottschalk,Substitution minimal sets, Trans. Amer. Math. Soc.109 (1963), 467–491.

    Article  MATH  MathSciNet  Google Scholar 

  5. K. Jacobs and M. Keane,0–1 Sequences of Toeplitz type, Z. Wahrscheinlichkeitstheorie Verw. Gebiete13 (1969), 123–131.

    Article  MATH  MathSciNet  Google Scholar 

  6. Teturo Kamae,Spectrum of a substitution minimal set, J. Math. Soc. Japan22 (1970), 567–578.

    Article  MATH  MathSciNet  Google Scholar 

  7. Teturo Kamae,A topological invariant of substitution minimal sets, J. Math. Soc. Japan24 (1972), 285–306.

    MATH  MathSciNet  Google Scholar 

  8. M. Keane,Generalized Morse sequences, Z. Wahrscheinlichkeitstheorie Verw. Gebiete10 (1968), 335–353.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. C. Oxtoby,Ergodic sets, Bull. Amer. Math. Soc.58 (1961), 116–136.

    Article  MathSciNet  Google Scholar 

  10. J. C. Oxtoby,On two theorems of Parthasarathy and Kakutani concerning the shift transformation, Ergodic Theory, Fred B. Wright (ed.), Academic Press, New York, 1962.

    Google Scholar 

  11. K. R. Parthasarathy,Probability Measures on Metric Spaces, Academic Press, New York, 1967.

    MATH  Google Scholar 

  12. V. A. Rokhlin,New progress in the theory of transformations with invariant measure, Russian Math. Surveys15 (1960), 1–22.

    Article  MATH  Google Scholar 

  13. H. Totoki,Introduction to Ergodic Theory, Kyoritsu Shuppan, Tokyo, 1971 (in Japanese).

    Google Scholar 

  14. B. Weiss,Normal sequences as collectives, Proc. Symp. on Topological Dynamics and Ergodic Theory, Univ. of Kentucky, 1971.

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Authors and Affiliations

  1. Department of Mathematics, Osaka City University, Sugimoto-cho, Osaka, Japan

    Teturo Kamae

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  1. Teturo Kamae
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Kamae, T. Subsequences of normal sequences. Israel J. Math. 16, 121–149 (1973). https://doi.org/10.1007/BF02757864

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  • DOI: https://doi.org/10.1007/BF02757864

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