Properties of Wiener-Wintner dynamical systems

I. Assani; K. Nicolaou

Bulletin de la Société Mathématique de France (2001)

  • Volume: 129, Issue: 3, page 361-377
  • ISSN: 0037-9484

Abstract

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In this paper we prove the following results. First, we show the existence of Wiener-Wintner dynamical system with continuous singular spectrum in the orthocomplement of their respective Kronecker factors. The second result states that if f ∈ L p , p large enough, is a Wiener-Wintner function then, for all γ ∈ ( 1 + 1 2 p - β 2 , 1 ] , there exists a set X f of full measure for which the series ∑ n = 1 ∞ f ( T n x ) e 2 π i n ϵ n γ converges uniformly with respect to ϵ .

How to cite

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Assani, I., and Nicolaou, K.. "Properties of Wiener-Wintner dynamical systems." Bulletin de la Société Mathématique de France 129.3 (2001): 361-377. <http://eudml.org/doc/272434>.

@article{Assani2001,
abstract = {In this paper we prove the following results. First, we show the existence of Wiener-Wintner dynamical system with continuous singular spectrum in the orthocomplement of their respective Kronecker factors. The second result states that if $f\in L^p$, $p$ large enough, is a Wiener-Wintner function then, for all $\gamma \in (1+\frac\{1\}\{2p\}-\frac\{\beta \}\{2\},1]$, there exists a set $X_f$ of full measure for which the series $\sum _\{n=1\}^\{\infty \} \frac\{f(T^n x)e^\{2\pi i n \epsilon \}\}\{n^\{\gamma \}\} $ converges uniformly with respect to $\epsilon $.},
author = {Assani, I., Nicolaou, K.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Wiener Wintner dynamical systems; Wiener Wintner functions; Kronecker factor},
language = {eng},
number = {3},
pages = {361-377},
publisher = {Société mathématique de France},
title = {Properties of Wiener-Wintner dynamical systems},
url = {http://eudml.org/doc/272434},
volume = {129},
year = {2001},
}

TY - JOUR
AU - Assani, I.
AU - Nicolaou, K.
TI - Properties of Wiener-Wintner dynamical systems
JO - Bulletin de la Société Mathématique de France
PY - 2001
PB - Société mathématique de France
VL - 129
IS - 3
SP - 361
EP - 377
AB - In this paper we prove the following results. First, we show the existence of Wiener-Wintner dynamical system with continuous singular spectrum in the orthocomplement of their respective Kronecker factors. The second result states that if $f\in L^p$, $p$ large enough, is a Wiener-Wintner function then, for all $\gamma \in (1+\frac{1}{2p}-\frac{\beta }{2},1]$, there exists a set $X_f$ of full measure for which the series $\sum _{n=1}^{\infty } \frac{f(T^n x)e^{2\pi i n \epsilon }}{n^{\gamma }} $ converges uniformly with respect to $\epsilon $.
LA - eng
KW - Wiener Wintner dynamical systems; Wiener Wintner functions; Kronecker factor
UR - http://eudml.org/doc/272434
ER -

References

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  1. [1] I. Assani – « Wiener-wintner dynamical systems », Preprint, 1998. Zbl1128.37300MR2032481
  2. [2] —, « Spectral characterization of Wiener-Wintner dynamical systems », Prépublication IRMA Strasbourg, June 2000. 
  3. [3] J. Bourgain – « Double recurrence and almost sure convergence », 404 (1990), p. 140–161. Zbl0685.28008MR1037434
  4. [4] H. Furstenberg – Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, NJ, 1981. Zbl0459.28023MR603625
  5. [5] A. Iwanik, M. Lemanczyk & C. Mauduit – « Spectral properties of piecewise absolutely continuous cocycles over irrational rotations », 59 (1996), p. 171–187. Zbl0931.28015MR1688497
  6. [6] A. Khinchin – « Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen », 92 (1924), p. 115–125. MR1512207JFM50.0125.01
  7. [7] —, Continued fractions, The University of Chicago press, Chicago, Illinois, 1964. Zbl0117.28601MR161833
  8. [8] L. Kuipers & H. Niederreiter – Uniform distribution of sequences, John Wiley and Sons, 1974. Zbl0281.10001MR419394
  9. [9] H. Medina – « Spectral Types of Unitary Operators Arising from Irrational Rotations on the Circle Group », 41 (1994), p. 39–49. Zbl0999.47024MR1260607
  10. [10] M. Schwartz – « Polynomially moving ergodic averages », 103 (1988), no. 1, p. 252–254. Zbl0642.28008MR938678

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