On a sequence of functions related to an iteration process. I, II. (Sur une suite de fonctions liées à un procédé itératif. I, II.) (French) Zbl 0828.11037

Publ. Dépt. Math. Limoges 12, 1-16 (1990); J. Théor. Nombres Bordx. 5, No. 2, 235-261 (1993).
The author considers the functions \(F_k = \beta F_{k - 1} + \alpha F_{k - 2} + (-1)^kf_0^{(k)}\) \((k \geq 2)\), \(F_0 = \text{id}\), \(F_1 = \beta \text{ id} - f_0\), where \(f_0(x) = \{\beta+ (3 + \beta) x\} - \beta\), \(\alpha = 2 - \sqrt 2\), \(\beta = \sqrt 2 - 1\); \(f_0^{(k)}\) denotes the \(k\)-th iteration and \(\{\cdot\}\) the fractional part. The main result gives a precise asymptotic formula for \(\sup F_k\) and \(\inf F_k\). The above iteration process is described via substitutions.
Reviewer: R.F.Tichy (Graz)


11K06 General theory of distribution modulo \(1\)
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