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Sums of continuous and differentiable functions in dynamical systems

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Abstract

LetT be a homeomorphism of a metrizable compactX, the sequencec k/k tends to 0 andc k tends to infinity. We’ll study the limit behaviour of the distributions of the sums (1/c k) ∑ i=0 k-1 F oT i whereF is from a space of continuous functions—the central limit problem and the speed of convergence in the ergodic theorem.

The main attention is given to the case whereX is the unit circle andT is an irrational rotation; in this case we consider the spaces of absolutely continuous, Lipschitz, andk-times differentiable functionsF.

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References

  1. [1]

    L. Baggett and K. Merrill,Smooth cocycles for an irrational rotation, Israel Journal of Mathematics79 (1992), 281–288.

  2. [2]

    R. Burton and M. Denker,On the central limit theorem for dynamical systems, Transactions of the American Mathematical Society302 (1987), 715–726.

  3. [3]

    I. P. Cornfeld, Ya. G. Sinai and S. V. Fomin,Ergodic Theory, Die Grundlehre der mathematischen Wissenschaften, 245, springer-Verlag, Berlin-Göttingen-Heidelberg, 1982.

  4. [4]

    T. De La Rue, S. Ladouceur, G. Peškir and M. Weber,On the central limit theorem for aperiodic systems and applications, preprint.

  5. [5]

    M. Denker, Ch. Grillenberger and K. Sigmund,Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics527, Springer-Verlag, Berlin-Heidelberg-New York, 1976.

  6. [6]

    P. Gabriel, M. Lemańczyk and P. Liardet,Ensemble d’invariants pour les produits croisés de Anzai, Mémoire de la Société Mathématique de France (novelle série) No 47—Suppl. au Bulletin de la S.M.F.119 (1991), 1–102.

  7. [7]

    R. Herman,Sur la conjugation différentiable des difféomorphismes du cercle à des rotations, Publications Mathématiques de l’Institut des Hautes Études Scientifiques49 (1979), 5–234.

  8. [8]

    A. B. Katok,Constructions in Ergodic Theory, manuscript.

  9. [9]

    M. Keane and D. Volný, preprint.

  10. [10]

    H. Kesten,On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arithmetica12 (1966), 193–212.

  11. [11]

    A. Ya. Khintchine,Continued Fractions, Noordhoff, Groningen, 1963.

  12. [12]

    U. Krengel,Ergodic Theorems (De Gruyter Studies in Mathematics 6), de Gruyter, Berlin, 1985.

  13. [13]

    L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Wiley, New York, 1974.

  14. [14]

    M. Lacey,On central limits theorems, modulus of continuity and diophantine type for irrational rotations, Journal d’Analyse Mathématique61 (1993), 47–59.

  15. [15]

    C. C. Moore and K. Schmidt,Coboundaries and homomorphisms for non-singular actions and a problem of H. Helson, Proceedings of the London Mathematical Society40 (1980), 443–475.

  16. [16]

    J. C. Oxtoby,Ergodic sets, Bulletin of the American Mathematical Society58 (1952), 116–136.

  17. [17]

    W. Parry and S. Tuncel,Classification Problems in Ergodic Theory, London Mathematical Society Lecture Notes No. 67, Cambridge University Press, Cambridge, 1982.

  18. [18]

    V. V. Petrov,Sums of Independent Random Variables, Springer-Verlag, Berlin, 1975.

  19. [19]

    D. VolnýOn limit theorems and catagory for dynamical systems, Yokohama Mathematical Journal38 (1990), 29–35.

  20. [20]

    D. Volný,Cohomology of Lipschitz and absolutely continuous functions for the circle rotation, submitted for publication.

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Author information

Correspondence to Pierre Liardet.

Additional information

The research of the first author was partially supported by URA-CNRS 225 and the research of the second author was partially supported by the Charles University grant GAUK 368.

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Liardet, P., Volný, D. Sums of continuous and differentiable functions in dynamical systems. Israel J. Math. 98, 29 (1997). https://doi.org/10.1007/BF02937328

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Keywords

  • Differentiable Function
  • Ergodic Theorem
  • Irrational Number
  • Continue Fraction Expansion
  • Irrational Rotation