Ergodic averages with deterministic weights

Fabien Durand[1]; Dominique Schneider[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)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 2, page 561-583
  • ISSN: 0373-0956

Abstract

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We study the convergence of the ergodic averages 1 N ∑ k = 0 N - 1 θ ( k ) f ∘ T u k where ( θ ( k ) ) k ∈ ℕ is a bounded sequence and ( u k ) k ∈ ℕ a strictly increasing sequence of integers such that Sup α ∈ ℝ | ∑ 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.

How to cite

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Durand, Fabien, and Schneider, Dominique. "Ergodic averages with deterministic weights." Annales de l’institut Fourier 52.2 (2002): 561-583. <http://eudml.org/doc/115987>.

@article{Durand2002,
abstract = {We study the convergence of the ergodic averages $\{1\over N\}\sum ^\{N-1\}_\{k=0\}\theta (k) f\circ T^\{u_k\}$ where $(\theta (k))_\{k\in \{\mathbb \{N\}\}\}$ is a bounded sequence and $(u_k)_\{k\in \{\mathbb \{N\}\}\}$ a strictly increasing sequence of integers such that $\{\rm Sup\}_\{\alpha \in \{\mathbb \{R\}\}\}\vert \sum ^\{N-1\}_\{k=0\}\theta (k)\{\rm exp\}(2i\pi \alpha u_k)\vert =O(N^\delta )$ for some $\delta &lt;1$. Moreover we give explicit such sequences $\theta $ and $u$ and we investigate in particular the case where $\theta $ is a $q$-multiplicative sequence.},
affiliation = {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); Université de Picardie Jules Verne, Faculté de Mathématiques & d’Informatique, Pôle de Saint-Leu, 33 rue Saint-Leu, 80039 Amiens Cedex 1 (France)},
author = {Durand, Fabien, Schneider, Dominique},
journal = {Annales de l’institut Fourier},
keywords = {weighted ergodic averages; central limit theorem; almost sure convergence; $q$-multiplicative sequences; substitutive sequences; generalized Thue-Morse sequences; -multiplicative sequences,substitution sequences},
language = {eng},
number = {2},
pages = {561-583},
publisher = {Association des Annales de l'Institut Fourier},
title = {Ergodic averages with deterministic weights},
url = {http://eudml.org/doc/115987},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Durand, Fabien
AU - Schneider, Dominique
TI - Ergodic averages with deterministic weights
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 2
SP - 561
EP - 583
AB - We study the convergence of the ergodic averages ${1\over N}\sum ^{N-1}_{k=0}\theta (k) f\circ T^{u_k}$ where $(\theta (k))_{k\in {\mathbb {N}}}$ is a bounded sequence and $(u_k)_{k\in {\mathbb {N}}}$ a strictly increasing sequence of integers such that ${\rm Sup}_{\alpha \in {\mathbb {R}}}\vert \sum ^{N-1}_{k=0}\theta (k){\rm exp}(2i\pi \alpha u_k)\vert =O(N^\delta )$ for some $\delta &lt;1$. Moreover we give explicit such sequences $\theta $ and $u$ and we investigate in particular the case where $\theta $ is a $q$-multiplicative sequence.
LA - eng
KW - weighted ergodic averages; central limit theorem; almost sure convergence; $q$-multiplicative sequences; substitutive sequences; generalized Thue-Morse sequences; -multiplicative sequences,substitution sequences
UR - http://eudml.org/doc/115987
ER -

References

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  1. J.-P. Allouche, M. Mendès France, On an extremal property of the Rudin-Shapiro sequence, Mathematika 32 (1985), 33-38 Zbl0561.10025MR817104
  2. J.-P. Allouche, P. Liardet, Generalized Rudin-Shapiro sequences, Acta Arith. 60 (1991), 1-27 Zbl0763.11010MR1129977
  3. J. Bourgain, Almost sure convergence and bounded entropy, Israel J. Math. 63 (1988), 79-97 Zbl0677.60042MR959049
  4. S. Durand, D. Schneider, Théorèmes ergodiques aléatoires et suite de poids régularisants, (2000) 
  5. A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith. 13 (1967-1968), 259-265 Zbl0155.09003MR220693
  6. N. Guillotin, D. Schneider, Ergodic theorems for dynamic random walks, (1999) Zbl0936.60027MR1710513
  7. R. L. Jones, M. Lin, J. Olsen, Weighted ergodic theorems along subsequences of density zero, New York J. Math 3A (1997-1998), 89-98 Zbl0906.47007MR1611121
  8. J.-P. Kahane, Some random series of functions, Cambridge Studies in Advanced Mathematics (1985) Zbl0571.60002MR833073
  9. M. Keane, Generalized Morse sequences, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10 (1968), 335-353 Zbl0162.07201MR239047
  10. U. Krengel, Ergodic theorems, 6 (1985), de Gruyter, Berlin-New York Zbl0575.28009MR797411
  11. L. Kuipers, H. Niederreiter, Uniform distribution of sequences, (1974), Wiley and Sons Zbl0281.10001MR419394
  12. M. T. Lacey, On central limit theorems, modulus of continuity and Diophantine type for irrational rotations, J. Anal. Math 61 (1993), 47-59 Zbl0790.60027MR1253438
  13. E. Lesigne, C. Mauduit, Propriétés ergodiques des suites q-multiplicatives, Compositio Math. 100 (1996), 131-169 Zbl0853.11064MR1383463
  14. E. Lesigne, C. Mauduit, B. Mossé, Le théorème ergodique le long d’une suite q -multiplicative, Compositio Math. 93 (1994), 49-79 Zbl0818.28006MR1286798
  15. M. Mendès France, Les suites à spectre vide et la répartition modulo 1, J. Number Theory 5 (1973), 1-15 Zbl0252.10033MR319909
  16. M. Mendès France, G. Tenenbaum, Dimension des courbes planes, papiers pliés et suites de Rudin-Shapiro, Bull. Soc. Math. France 109 (1981), 207-215 Zbl0468.10033MR623789
  17. H. M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc 22 (1921), 84-100 Zbl48.0786.06MR1501161
  18. M. Queffélec, Substitution dynamical systems--spectral analysis, 1294 (1987), Springer-Verlag, Berlin Zbl0642.28013MR924156
  19. W. Rudin, Some theorems on Fourier coefficients, Proc. Amer. Math. Soc 10 (1959), 855-859 Zbl0091.05706MR116184
  20. D. Schneider, M. Weber, Weighted averages of contractions along subsequences, Convergence in ergodic theory and probability (Columbus, OH, 1993) 5 (1996), 397-404, Ohio State Univ. Math. Res. Inst. Publ, de Gruyter, Berlin Zbl0864.60024
  21. H. Shapiro, Extremal problems for polynomials and power series, (1952) 

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