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On Hartman Uniform Distribution and Measures on Compact Spaces

Part of the Trends in Mathematics book series (TM)

Abstract

Abstract

Given a sequence of natural numbers
$$\mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 1}^N {{e^{2\pi i{k_n}x}} = 0}$$
we say it is Hartman uniformly distributed if
$$\mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 1}^N {f\left( {{T^{{k_n}}}x} \right) = } {e_{T,f}}\left( x \right)$$
for every non-integer real number x. This property of k being Hartman uniform distributed is interesting in subsequence ergodic theory because if for somepin [1,2], for a functionf ∈ L P (X, ß, μ) we have the limitn=1 existing almost everywhere with respect to the measure μ, then just the ergodicity of the dynamical system (X, ß, μ,T)implies that 1 t,f (x) =? x fdμ, which is of course useful with regard to applications. Not every sequence for which a pointwise convergence theorem holds has this property. In this paper we use this observation to give some results about invariant measures for continuous maps of compact metric spaces.

Keywords

Natural Number Invariant Measure Compact Space Ergodic Theorem Dominate Convergence Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AN]
    N. H. Asmar and R. Nair: Certain averages on the a-adic numbers, Proc. Amer. Math. Soc. 114 no. 1 (1992), 21–28.MathSciNetGoogle Scholar
  2. [Bol]
    J. Bourgain: On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), 39–72.MathSciNetMATHCrossRefGoogle Scholar
  3. [Bo2]
    J. Bourgain: Pointwise ergodic theorems for arithmetic sets, Publ. I. H. E. S. 69 (1989), 5–45.MathSciNetMATHGoogle Scholar
  4. [H]
    S. Hartman: Remarks on equidistribution on non-compact groups, Compositio Math., 16 (1964), 66–71.MathSciNetMATHGoogle Scholar
  5. [Ka]
    Y. Katznelson: An introduction to Harmonic Analysis, Wiley, (1968).MATHGoogle Scholar
  6. [Kr]
    U. Krengel:Ergodic Theorems, de Gruyter Studies in Mathematics 6 (1985).Google Scholar
  7. [KN]
    L. Kuipers and H. Neiderreiter: Uniform Distribution of Sequences, Wiley, (1974).MATHGoogle Scholar
  8. [Nal]
    R. Nair: On polynomials in primes and J. Bourgains circle method approach to ergodic theorems, Ergododic Theory and Dynamical Systems,11(1991), 485–499.MATHGoogle Scholar
  9. [Na2]
    R. Nair: On polynomials in primes and J. Bourgains circle method approach to ergodic theorems II, Studia Math.105(3) (1993), 207–233.MathSciNetMATHGoogle Scholar
  10. [Na3]
    R. Nair:On uniformly distributed sequences of integers and recurrence, (preprint).Google Scholar
  11. [Na4]
    R. Nair:On uniformly distributed sequences of integers and recurrence II, (preprint).Google Scholar
  12. [N]
    I. Niven: Uniform distribution of sequences of integers, Trans. Amer. Math. Soc. 98 (1961), 52–61.MathSciNetMATHCrossRefGoogle Scholar
  13. [W]
    R. Walters: An Introduction to Ergodic Theory, Springer-Verlag, Graduate Texts in Mathematics 79 (1981).Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • R. Nair
    • 1
  1. 1.Department of Pure MathematicsUniversity of LiverpoolLiverpoolUK

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