Hofstadter's $G$ function is recursively defined via $G(0)=0$ and then $G(n)=n-G(G(n-1))$. Following Hofstadter, a family $(F_k)$ of similar functions is obtained by varying the number $k$ of nested recursive calls in this equation. We establish here that this family is ordered pointwise: for all $k$ and $n$, $F_k(n) \le F_{k+1}(n)$. For achieving this, a detour is made via infinite morphic words generalizing the Fibonacci word. Various properties of these words are proved, concerning the lengths of substituted prefixes of these words and the counts of some specific letters in these prefixes. We also relate the limits of $\frac{1}{n}F_k(n)$ to the frequencies of letters in the considered words.

翻译：霍夫施塔特函数 $G$ 通过 $G(0)=0$ 及 $G(n)=n-G(G(n-1))$ 递归定义。继霍夫施塔特之后，通过改变该方程中嵌套递归调用的次数 $k$，可得到一类相似函数族 $(F_k)$。本文证明该函数族具有点态序关系：对所有 $k$ 和 $n$，均有 $F_k(n) \le F_{k+1}(n)$。为达成此证明，我们通过推广斐波那契词的无限态射词进行迂回论证。证明了这些词的若干性质，包括其前缀替换长度的特性以及这些前缀中特定字母的计数规律。此外，我们将 $\frac{1}{n}F_k(n)$ 的极限与所研究词中字母的频率联系起来。