Ergodic averages with deterministic weights

Durand, Fabien; Schneider, Dominique

doi:10.5802/aif.1894

Ergodic averages with deterministic weights [ Moyennes ergodiques pondérées par des poids déterministes ]

Durand, Fabien ; Schneider, Dominique

Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 561-583.

Résumé
Abstract

Nous étudions la convergence de moyennes ergodiques $\frac{1}{N} \sum_{k = 0}^{N - 1} θ (k) f \circ T^{u_{k}}$ où $(θ (k))_{k \in ℕ}$ est une suite bornée et $(u_{k})_{k \in ℕ}$ une suite d’entiers strictement croissante tels que ${Sup}_{α \in ℝ} | \sum_{k = 0}^{N - 1} θ (k) \exp (2 i π α u_{k}) | = O (N^{δ})$ 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.

We study the convergence of the ergodic averages $\frac{1}{N} \sum_{k = 0}^{N - 1} θ (k) f \circ T^{u_{k}}$ where $(θ (k))_{k \in ℕ}$ is a bounded sequence and $(u_{k})_{k \in ℕ}$ a strictly increasing sequence of integers such that ${Sup}_{α \in ℝ} | \sum_{k = 0}^{N - 1} θ (k) \exp (2 i π α u_{k}) | = O (N^{δ})$ 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.

MR 1906483 | Zbl 1054.37003

DOI : https://doi.org/10.5802/aif.1894
Classification : 37A05, 28D05, 11K99
Mots clés: moyennes ergodiques pondérées, théorème ..., convergence presque sûre, suites multiplicatives

q

, suites substitutives, suite de Thue-Morse généralisées

@article{AIF_2002__52_2_561_0,
     author = {Durand, Fabien and Schneider, Dominique},
     title = {Ergodic averages with deterministic weights},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {2},
     year = {2002},
     pages = {561-583},
     doi = {10.5802/aif.1894},
     mrnumber = {1906483},
     zbl = {1054.37003},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2002__52_2_561_0/}
}

Durand, Fabien; Schneider, Dominique. Ergodic averages with deterministic weights. Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 561-583. doi : 10.5802/aif.1894. https://aif.centre-mersenne.org/item/AIF_2002__52_2_561_0/

Bibliographie

[AL] J.-P. Allouche; P. Liardet Generalized Rudin-Shapiro sequences, Acta Arith., Tome 60 (1991), pp. 1-27 | | MR 1129977 | Zbl 0763.11010

[AM] J.-P. Allouche; M. Mendès France On an extremal property of the Rudin-Shapiro sequence, Mathematika, Tome 32 (1985), pp. 33-38 | Article | MR 817104 | Zbl 0561.10025

[Bo] J. Bourgain Almost sure convergence and bounded entropy, Israel J. Math., Tome 63 (1988), pp. 79-97 | Article | MR 959049 | Zbl 0677.60042

[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., Tome 13 (1967-1968), pp. 259-265 | | MR 220693 | Zbl 0155.09003

[GS] N. Guillotin; D. Schneider Ergodic theorems for dynamic random walks (1999) (preprint LAMFA (Université de Picardie Jules Verne)) | MR 1710513 | Zbl 0936.60027

[JLO] R. L. Jones; M. Lin; J. Olsen Weighted ergodic theorems along subsequences of density zero, New York J. Math, Tome 3A (1997-1998), pp. 89-98 | | MR 1611121 | Zbl 0906.47007

[Ka] J.-P. Kahane Some random series of functions, Cambridge Studies in Advanced Mathematics (1985) | MR 833073 | Zbl 0571.60002

[Ke] M. Keane Generalized Morse sequences, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Tome 10 (1968), pp. 335-353 | Article | MR 239047 | Zbl 0162.07201

[KN] L. Kuipers; H. Niederreiter Uniform distribution of sequences, Wiley and Sons, 1974 | MR 419394 | Zbl 0281.10001

[Kr] U. Krengel Ergodic theorems, Studies in Mathematics, Tome 6, de Gruyter, Berlin-New York, 1985 | MR 797411 | Zbl 0575.28009

[La] M. T. Lacey On central limit theorems, modulus of continuity and Diophantine type for irrational rotations, J. Anal. Math, Tome 61 (1993), pp. 47-59 | Article | MR 1253438 | Zbl 0790.60027

[LM] E. Lesigne; C. Mauduit Propriétés ergodiques des suites q-multiplicatives, Compositio Math., Tome 100 (1996), pp. 131-169 | Numdam | MR 1383463 | Zbl 0853.11064

[LMM] E. Lesigne; C. Mauduit; B. Mossé Le théorème ergodique le long d'une suite $q$-multiplicative, Compositio Math., Tome 93 (1994), pp. 49-79 | Numdam | MR 1286798 | Zbl 0818.28006

[M] M. Mendès France Les suites à spectre vide et la répartition modulo 1, J. Number Theory, Tome 5 (1973), pp. 1-15 | Article | MR 319909 | Zbl 0252.10033

[Mo] H. M. Morse Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc, Tome 22 (1921), pp. 84-100 | Article | JFM 48.0786.06 | MR 1501161

[MT] M. Mendès France; G. Tenenbaum Dimension des courbes planes, papiers pliés et suites de Rudin-Shapiro, Bull. Soc. Math. France, Tome 109 (1981) no. 2, pp. 207-215 | Numdam | MR 623789 | Zbl 0468.10033

[Qu] M. Queffélec Substitution dynamical systems--spectral analysis, Lecture Notes in Mathematics, Tome 1294, Springer-Verlag, Berlin, 1987 | MR 924156 | Zbl 0642.28013

[Ru] W. Rudin Some theorems on Fourier coefficients, Proc. Amer. Math. Soc, Tome 10 (1959), pp. 855-859 | Article | MR 116184 | Zbl 0091.05706

[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), Tome 5 (1996), pp. 397-404 | Zbl 0864.60024

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