On functions with bounded remainder

Hellekalek, P.; Larcher, Gerhard

doi:10.5802/aif.1156

Hellekalek, P. ; Larcher, Gerhard

Annales de l'Institut Fourier, Tome 39 (1989) no. 1, pp. 17-26.

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 $ℝ / ℤ \times ℝ ?$

MR Zbl

DOI : 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},
     pages = {17--26},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {1},
     year = {1989},
     doi = {10.5802/aif.1156},
     mrnumber = {90i:28024},
     zbl = {0674.28007},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1156/}
}

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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. http://www.numdam.org/articles/10.5802/aif.1156/

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