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

Borel, Jean-Pierre

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)

MSC:

11K06

General theory of distribution modulo \(1\)

Keywords:

asymptotic formula; iteration process; substitutions

References:

[1]	Bertrand-Mathis, A., Développements en base θ ; répartition de la suite (xθn) ; langages codés et θ-shift, Bull. Soc. math. France114 (1986), 271-323. · Zbl 0628.58024
[2]	Borel, J.-P., Contribution à l’étude de la distribution des suites, Thèse d’État, Marseille-Luminy, 1989.
[3]	Borel, J.-P., Sur une suite de fonctions liées à un processus itératif, Publications du Département de Mathématiques deLimoges12 (1990), 1-16. · Zbl 0828.11037
[4]	Borel, J.-P., Symbolic representation of piecewise linear functions on the unit interval, Theoret. Comput. Sci. A 123 (1994), à paraître. · Zbl 0813.11045
[5]	Dupain, Y., Discrépance à l’origine de la suite (n(1+√5)/2), Ann. Inst. Fourier (Grenoble) 29 (1979), 81-106. · Zbl 0386.10021
[6]	Haber, S., On a sequence of points of interest for numerical quadrature, Res. Nat. Bur. Standards Sect. B70 (1966), 127-134. · Zbl 0158.16002
[7]	Hofbauer, F., On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math.34 (1979), 213-237. · Zbl 0422.28015
[8]	Hofbauer, F., The maximal measure for linear mod one transformations, J. London Math. Soc.23 (1981), 92-112. · Zbl 0431.54025
[9]	Parry, W., On the β-expansion of real numbers, Acta Math. Acad. Sci. Hungar.11 (1960), 401-416. · Zbl 0099.28103
[10]	Renyi, A., Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar8 (1957), 477-493. · Zbl 0079.08901