On a sequence of functions related to an iteration process. I, II (Q1320514)

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.