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,

φ n ( x ) : = φ ( x ) + φ ( T x ) + ⋯ + φ ( T n - 1 x ) .

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

φ n ( x ) : = φ ( x ) + φ ( T x ) + ... + φ ( T n - 1 x ) , 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?

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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://archive.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://archive.numdam.org/articles/10.5802/aif.1156/

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