Properties of Wiener-Wintner dynamical systems
Bulletin de la Société Mathématique de France, Volume 129 (2001) no. 3, pp. 361-377.

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 fL p , p large enough, is a Wiener-Wintner function then, for all γ(1+1 2p-β 2,1], there exists a set X f of full measure for which the series n=1 f(T n x)e 2πinϵ n γ converges uniformly with respect to ϵ.

Dans cette note nous démontrons les résultats suivants. Tout d’abord nous montrons l’existence de systèmes dynamiques ergodiques du type Wiener Wintner ayant un spectre singulier continu dans l’orthogonal de leur facteurs de Kronecker.Ensuite nous montrons que si fL p est une fonction du type Wiener-Wintner alors, pour γ(1+1 2p-β 2,1] on peut trouver un ensemble X f de mesure pleine pour lequel la série n=1 f(T n x)e 2πinϵ n γ converge uniformément en ϵ.

DOI: 10.24033/bsmf.2402
Classification: 28D05,  11K38
Keywords: Wiener Wintner dynamical systems, Wiener Wintner functions, Kronecker factor
@article{BSMF_2001__129_3_361_0,
     author = {Assani, I. and Nicolaou, K.},
     title = {Properties of {Wiener-Wintner} dynamical systems},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {361--377},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {129},
     number = {3},
     year = {2001},
     doi = {10.24033/bsmf.2402},
     zbl = {0994.37004},
     mrnumber = {1881201},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/bsmf.2402/}
}
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
DA  - 2001///
SP  - 361
EP  - 377
VL  - 129
IS  - 3
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/bsmf.2402/
UR  - https://zbmath.org/?q=an%3A0994.37004
UR  - https://www.ams.org/mathscinet-getitem?mr=1881201
UR  - https://doi.org/10.24033/bsmf.2402
DO  - 10.24033/bsmf.2402
LA  - en
ID  - BSMF_2001__129_3_361_0
ER  - 
%0 Journal Article
%A Assani, I.
%A Nicolaou, K.
%T Properties of Wiener-Wintner dynamical systems
%J Bulletin de la Société Mathématique de France
%D 2001
%P 361-377
%V 129
%N 3
%I Société mathématique de France
%U https://doi.org/10.24033/bsmf.2402
%R 10.24033/bsmf.2402
%G en
%F BSMF_2001__129_3_361_0
Assani, I.; Nicolaou, K. Properties of Wiener-Wintner dynamical systems. Bulletin de la Société Mathématique de France, Volume 129 (2001) no. 3, pp. 361-377. doi : 10.24033/bsmf.2402. http://www.numdam.org/articles/10.24033/bsmf.2402/

[1] I. Assani - « Wiener-wintner dynamical systems », Preprint, 1998. | MR | Zbl

[2] -, « Spectral characterization of Wiener-Wintner dynamical systems », Prépublication IRMA Strasbourg, June 2000.

[3] J. Bourgain - « Double recurrence and almost sure convergence », 404 (1990), p. 140-161. | MR | Zbl

[4] H. Furstenberg - Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, NJ, 1981. | MR | Zbl

[5] A. Iwanik, M. Lemanczyk & C. Mauduit - « Spectral properties of piecewise absolutely continuous cocycles over irrational rotations », 59 (1996), p. 171-187. | MR | Zbl

[6] A. Khinchin - « Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen », 92 (1924), p. 115-125. | JFM | MR

[7] -, Continued fractions, The University of Chicago press, Chicago, Illinois, 1964. | MR | Zbl

[8] L. Kuipers & H. Niederreiter - Uniform distribution of sequences, John Wiley and Sons, 1974. | MR | Zbl

[9] H. Medina - « Spectral Types of Unitary Operators Arising from Irrational Rotations on the Circle Group », 41 (1994), p. 39-49. | MR | Zbl

[10] M. Schwartz - « Polynomially moving ergodic averages », 103 (1988), no. 1, p. 252-254. | MR | Zbl

Cited by Sources: