On functions with bounded remainder

Hellekalek, P.; Larcher, Gerhard

Annales de l'Institut Fourier, Tome 39 (1989), p. 17-26 / Harvested from Numdam

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Résumé
Abstract

Soit $T : ℝ / ℤ \to ℝ / ℤ$ une transformation du type Neumann-Kakutani en base $q$ et soit $φ \in C^{1} ([0, 1])$ . Posons, pour $x \in ℝ / ℤ$ , $n \in ℕ$ ,

φ_{n} (x) : = φ (x) + φ (T x) + \dots + φ (T^{n - 1} x) .

Nous étudions les trois questions suivantes :

1. Pour la suite $(φ_{n} (x))_{n \geq 1}$ : à quelles conditions sera-t-elle bornée ?

2. Que peut-on dire sur les points d’adhérence de $(φ_{n} (x))_{n \geq 1} ?$

3. Pour le produit croisé $(x, y) \mapsto (T x, y + φ (x))$ sur le cylindre $ℝ / ℤ \times ℝ$ : à quelles conditions sera-t-il ergodique ?

Let $T : ℝ / ℤ \to ℝ / ℤ$ be a von Neumann-Kakutani $q$ - adic adding machine transformation and let $φ \in C^{1} ([0, 1])$ . Put

φ_{n} (x) : = φ (x) + φ (T x) + ... + φ (T^{n - 1} x), x \in ℝ / ℤ, n \in ℕ .

We study three questions:

1. When will $(φ_{n} (x))_{n \geq 1}$ be bounded?

2. What can be said about limit points of $(φ_{n} (x))_{n \geq 1} ?$

3. When will the skew product $(x, y) \mapsto (T x, y + φ (x))$ be ergodic on $ℝ / ℤ \times ℝ ?$

Publié le : 1989-01-01

MR 90i:28024 | Zbl 0674.28007

DOI : https://doi.org/10.5802/aif.1156

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     author = {Hellekalek, P. and Larcher, Gerhard},
     title = {On functions with bounded remainder},
     journal = {Annales de l'Institut Fourier},
     volume = {39},
     year = {1989},
     pages = {17-26},
     doi = {10.5802/aif.1156},
     mrnumber = {90i:28024},
     zbl = {0674.28007},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1989__39_1_17_0}
}

Hellekalek, P.; Larcher, Gerhard. On functions with bounded remainder. Annales de l'Institut Fourier, Tome 39 (1989) pp. 17-26. doi : 10.5802/aif.1156. http://gdmltest.u-ga.fr/item/AIF_1989__39_1_17_0/

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