no. 1
On functions with bounded remainder
Annales de l'Institut Fourier, Tome 39 (1989) no. 1, pp. 17-26.

Soit T:ℝ/ℤ→ℝ/ℤ une transformation du type Neumann-Kakutani en base q et soit φ∈C 1 ([0,1]). Posons, pour x∈ℝ/ℤ, n∈ℕ,

φn(x):=φ(x)+φ(Tx)+⋯+φ(Tn-1x).

Nous étudions les trois questions suivantes :

1. Pour la suite (φ n (x)) n≥1 : à quelles conditions sera-t-elle bornée ?

2. Que peut-on dire sur les points d’adhérence de (φ n (x)) n≥1 ?

3. Pour le produit croisé (x,y)↦(Tx,y+φ(x)) sur le cylindre ℝ/ℤ×ℝ : à quelles conditions sera-t-il ergodique ?

Let T:ℝ/ℤ→ℝ/ℤ be a von Neumann-Kakutani q- adic adding machine transformation and let φ∈C 1 ([0,1]). Put

φn(x):=φ(x)+φ(Tx)+...+φ(Tn-1x),x∈ℝ/ℤ,n∈ℕ.

We study three questions:

1. When will (φ n (x)) n≥1 be bounded?

2. What can be said about limit points of (φ n (x)) n≥1 ?

3. When will the skew product (x,y)↦(Tx,y+φ(x)) be ergodic on ℝ/ℤ×ℝ?

@article{AIF_1989__39_1_17_0,
     author = {Hellekalek, P. and Larcher, Gerhard},
     title = {On functions with bounded remainder},
     journal = {Annales de l'Institut Fourier},
     pages = {17--26},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {1},
     year = {1989},
     doi = {10.5802/aif.1156},
     zbl = {0674.28007},
     mrnumber = {90i:28024},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1156/}
}
TY  - JOUR
AU  - Hellekalek, P.
AU  - Larcher, Gerhard
TI  - On functions with bounded remainder
JO  - Annales de l'Institut Fourier
PY  - 1989
SP  - 17
EP  - 26
VL  - 39
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1156/
DO  - 10.5802/aif.1156
LA  - en
ID  - AIF_1989__39_1_17_0
ER  - 
%0 Journal Article
%A Hellekalek, P.
%A Larcher, Gerhard
%T On functions with bounded remainder
%J Annales de l'Institut Fourier
%D 1989
%P 17-26
%V 39
%N 1
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1156/
%R 10.5802/aif.1156
%G en
%F AIF_1989__39_1_17_0
Hellekalek, P.; Larcher, Gerhard. On functions with bounded remainder. Annales de l'Institut Fourier, Tome 39 (1989) no. 1, pp. 17-26. doi : 10.5802/aif.1156. https://aif.centre-mersenne.org/articles/10.5802/aif.1156/

[1] Y. Dupain and V.T. Sós, On the one-sided boundedness of discrepancy-function of the sequence {nα}, Acta Arith., 37 (1980), 363-374. | MR | Zbl

[2] H. Faure, Etude des restes pour les suites de Van der Corput généralisées, J. Number Th., 16 (1983), 376-394. | MR | Zbl

[3] W.H. Gottschalk and G.A. Hedlund, Topological Dynamics, AMS Colloq. Publ., 1955. | MR | Zbl

[4] P. Hellekalek, Regularities in the distribution of special sequences, J. Number Th., 18 (1984), 41-55. | MR | Zbl

[5] P. Hellekalek, Ergodicity of a class of cylinder flows related to irregularities of distribution, Comp. Math., 61 (1987), 129-136. | Numdam | MR | Zbl

[6] P. Hellekalek and G. Larcher, On the ergodicity of a class of skew products, Israel J. Math., 54 (1986), 301-306. | MR | Zbl

[7] L.K. Hua and Y. Wang, Applications of number theory to numerical analysis, Springer-Verlag, Berlin, New York, 1981. | MR | Zbl

[8] H. Kesten, On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arith., 12 (1966), 193-212. | MR | Zbl

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

[10] I. Oren, Ergodicity of cylinder flows arising from irregularities of distribution, Israel J. Math., 44 (1983), 127-138. | MR | Zbl

[11] K. Petersen, On a series of cosecants related to a problem in ergodic theory, Comp. Math., 26 (1973), 313-317. | Numdam | MR | Zbl

Cité par Sources :