Ergodic averages with deterministic weights
Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 561-583.

We study the convergence of the ergodic averages 1 N k=0 N-1 θ(k)fT u k where (θ(k)) k is a bounded sequence and (u k ) k a strictly increasing sequence of integers such that Sup α | k=0 N-1 θ(k) exp (2iπαu k )|=O(N δ ) for some δ<1. Moreover we give explicit such sequences θ and u and we investigate in particular the case where θ is a q-multiplicative sequence.

Nous étudions la convergence de moyennes ergodiques 1 N k=0 N-1 θ(k)fT u k (θ(k)) k est une suite bornée et (u k ) k une suite d’entiers strictement croissante tels que Sup α | k=0 N-1 θ(k) exp (2iπαu k )|=O(N δ ) avec δ<1. De plus nous donnons des exemples explicites de telles suites θ et u puis, nous nous intéressons aux cas où θ est une suite q- multiplicative.

DOI: 10.5802/aif.1894
Classification: 37A05, 28D05, 11K99
Keywords: weighted ergodic averages, central limit theorem, almost sure convergence, $q$-multiplicative sequences, substitutive sequences, generalized Thue-Morse sequences
Mot clés : moyennes ergodiques pondérées, théorème ..., convergence presque sûre, suites multiplicatives $q$, suites substitutives, suite de Thue-Morse généralisées
Durand, Fabien 1; Schneider, Dominique 2

1 Universidad de Chile, Centro de Modelamiento Matemático, Casilla 170-3, Correo 3, Santiago (chili) \& Université de Picardie Jules Verne, Faculté de Mathématiques \& d'Informatique, Pôle de Saint-Leu, 33 rue Saint-Leu, 80039 Amiens Cedex 1 (France)
2 Université de Picardie Jules Verne, Faculté de Mathématiques \& d'Informatique, Pôle de Saint-Leu, 33 rue Saint-Leu, 80039 Amiens Cedex 1 (France)
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Durand, Fabien; Schneider, Dominique. Ergodic averages with deterministic weights. Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 561-583. doi : 10.5802/aif.1894. http://archive.numdam.org/articles/10.5802/aif.1894/

[AL] J.-P. Allouche; P. Liardet Generalized Rudin-Shapiro sequences, Acta Arith., Volume 60 (1991), pp. 1-27 | EuDML | MR | Zbl

[AM] J.-P. Allouche; M. Mendès France On an extremal property of the Rudin-Shapiro sequence, Mathematika, Volume 32 (1985), pp. 33-38 | DOI | MR | Zbl

[Bo] J. Bourgain Almost sure convergence and bounded entropy, Israel J. Math., Volume 63 (1988), pp. 79-97 | DOI | MR | Zbl

[DS] S. Durand; D. Schneider Théorèmes ergodiques aléatoires et suite de poids régularisants (2000) Preprint LAMFA (Université de Picardie Jules Verne)

[G] A. O. Gel'fond Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., Volume 13 (1967-1968), pp. 259-265 | EuDML | MR | Zbl

[GS] N. Guillotin; D. Schneider Ergodic theorems for dynamic random walks (1999) preprint LAMFA (Université de Picardie Jules Verne) | MR | Zbl

[JLO] R. L. Jones; M. Lin; J. Olsen Weighted ergodic theorems along subsequences of density zero, New York J. Math, Volume 3A (1997-1998), pp. 89-98 | EuDML | MR | Zbl

[Ka] J.-P. Kahane Some random series of functions, Cambridge Studies in Advanced Mathematics (1985) | MR | Zbl

[Ke] M. Keane Generalized Morse sequences, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Volume 10 (1968), pp. 335-353 | DOI | MR | Zbl

[KN] L. Kuipers; H. Niederreiter Uniform distribution of sequences, Wiley and Sons, 1974 | MR | Zbl

[Kr] U. Krengel Ergodic theorems, Studies in Mathematics, 6, de Gruyter, Berlin-New York, 1985 | MR | Zbl

[La] M. T. Lacey On central limit theorems, modulus of continuity and Diophantine type for irrational rotations, J. Anal. Math, Volume 61 (1993), pp. 47-59 | DOI | MR | Zbl

[LM] E. Lesigne; C. Mauduit Propriétés ergodiques des suites q-multiplicatives, Compositio Math., Volume 100 (1996), pp. 131-169 | Numdam | MR | Zbl

[LMM] E. Lesigne; C. Mauduit; B. Mossé Le théorème ergodique le long d'une suite q-multiplicative, Compositio Math., Volume 93 (1994), pp. 49-79 | Numdam | MR | Zbl

[M] M. Mendès France Les suites à spectre vide et la répartition modulo 1, J. Number Theory, Volume 5 (1973), pp. 1-15 | DOI | MR | Zbl

[Mo] H. M. Morse Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc, Volume 22 (1921), pp. 84-100 | DOI | JFM | MR

[MT] M. Mendès France; G. Tenenbaum Dimension des courbes planes, papiers pliés et suites de Rudin-Shapiro, Bull. Soc. Math. France, Volume 109 (1981) no. 2, pp. 207-215 | Numdam | MR | Zbl

[Qu] M. Queffélec Substitution dynamical systems--spectral analysis, Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 1987 | MR | Zbl

[Ru] W. Rudin Some theorems on Fourier coefficients, Proc. Amer. Math. Soc, Volume 10 (1959), pp. 855-859 | DOI | MR | Zbl

[Sh] H. Shapiro Extremal problems for polynomials and power series (1952) (Thesis, M.I.T)

[SW] D. Schneider; M. Weber Weighted averages of contractions along subsequences, Convergence in ergodic theory and probability (Columbus, OH, 1993), Volume 5 (1996), pp. 397-404 | Zbl

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