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Subdiffusive behavior generated by irrational rotations

Published online by Cambridge University Press:  01 August 2009

F. HUVENEERS
F. HUVENEERS*
Affiliation:
UcL, FYMA, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium (email: francois.huveneers@uclouvain.be)
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Abstract

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We study asymptotic distributions of the sums yn(x)=∑ k=0n−1ψ(x+kα) with respect to the Lebesgue measure, where α∈ℝ−ℚ and where ψ is the 1-periodic function of bounded variation such that ψ(x)=1 if x∈[0,1/2[ and ψ(x)=−1 if x∈[1/2,1[. For every α∈ℝ−ℚ, we find a sequence (nj)j⊂ℕ such that $y_{n_j}/\sqrt j$ is asymptotically normally distributed. For n≥1, let zn∈(ym)m≤n be such that ‖zn‖L2=max m≤n‖ym‖L2. If α is of constant type, we show that zn/‖zn‖L2 is also asymptotically normally distributed. We give a heuristic link with the theory of expanding maps of the interval.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1217 - 1233
Copyright
Copyright © Cambridge University Press 2009

References

[1]Adamczewski, B.. Répartition des suites (nα)n∈ℕ et substitutions. Acta Arith. 112 (2004), 1–22.CrossRefGoogle Scholar
[2]Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, River Edge, NJ, 2000.CrossRefGoogle Scholar
[3]Bricmont, J., Gawedzki, K. and Kupiainen, A.. KAM theorem and quantum field theory. Comm. Math. Phys. 201(3) (1999), 699–727.CrossRefGoogle Scholar
[4]Bunimovich, L. A. and Sinai, Ya. G.. Statistical properties of Lorentz gas with a periodic configuration of scatterers. Comm. Math. Phys. 72 (1981), 479–497.CrossRefGoogle Scholar
[5]Burton, R. and Denker, M.. On the central limit theorem for dynamical systems. Trans. Amer. Math. Soc. 302(2) (1987), 715–726.CrossRefGoogle Scholar
[6]De La Rue, T., Ladouceur, S., Peskir, G. and Weber, M.. On the central limit theorem for aperiodic dynamical systems and applications. Theory Probab. Math. Statist. 57 (1997), 140–159.Google Scholar
[7]Dettmann, C. P. and Cohen, E. G. D.. Microscopic chaos and diffusion. J. Stat. Phys. 101(3-4) (2000), 775–817.CrossRefGoogle Scholar
[8]Drmota, M. and Tichy, R. F.. Sequences, Discrepancies and Applications (Lecture Notes in Mathematics, 1651). Springer, Berlin, 1997.CrossRefGoogle Scholar
[9]Gaspard, P., Briggs, M. E., Francis, M. K., Sengers, J. V., Gammon, R. W., Dorfman, J. R. and Calabrese, R. V.. Experimental evidence for microscopic chaos. Nature 394 (1998), 865–868.CrossRefGoogle Scholar
[10]Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers, 4th edn. Oxford University Press, Oxford, 1960.Google Scholar
[11]Herman, M. R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5–233.CrossRefGoogle Scholar
[12]Hofbauer, F. and Keller, G.. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982), 119–140.CrossRefGoogle Scholar
[13]Kuipers, L. and Niederreiter, H.. Uniform Distribution of Sequences. Wiley, New York, 1974.Google Scholar
[14]Liardet, P. and Volný, D.. Sums of continuous and differentiable functions in dynamical systems. Israel J. Math. 98 (1997), 29–60.CrossRefGoogle Scholar