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

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

 ${\phi }_{n}\left(x\right):=\phi \left(x\right)+\phi \left(Tx\right)+...+\phi \left({T}^{n-1}x\right),\phantom{\rule{4pt}{0ex}}x\in ℝ/ℤ,\phantom{\rule{4pt}{0ex}}n\in ℕ.$

We study three questions:

1. When will $\left({\phi }_{n}\left(x\right){\right)}_{n\ge 1}$ be bounded?

2. What can be said about limit points of $\left({\phi }_{n}\left(x\right){\right)}_{n\ge 1}?$

3. When will the skew product $\left(x,y\right)↦\left(Tx,y+\phi \left(x\right)\right)$ be ergodic on $ℝ/ℤ×ℝ?$

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

 ${\phi }_{n}\left(x\right):=\phi \left(x\right)+\phi \left(Tx\right)+\cdots +\phi \left({T}^{n-1}x\right).$

Nous étudions les trois questions suivantes :

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

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

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

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title = {On functions with bounded remainder},
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publisher = {Institut Fourier},
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Hellekalek, P.; Larcher, Gerhard. On functions with bounded remainder. Annales de l'Institut Fourier, Volume 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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