On the Iterates of Digit Maps

Abstract

Given a base $b$, a "digit map" is a map $f:{\mathbb{Z}}^{\ge 0}\to {\mathbb{Z}}^{\ge 0}$ of the form $f({\sum }_{i=0}^n{a}_i{b}^i)={\sum }_{i=0}^n{f}_{\ast }(a_i)$, $0\le a_i\le b-1$ for each $i$, where $f_{\ast }:\{0,1,\dots ,b-1\}\to {\mathbb{Z}}^{\ge 0}$ satisfies $f_{\ast }(0)=0$ and $f_{\ast }(1)=1$. It has been proven for $b=10$ and $f_{\ast }(m)=m^2$, and various generalizations thereof, that there are arbitrarily long sequences of consecutive positive integers that end up at $1$ under repeated application of $f$. In this paper, we significantly generalize these results, providing a complete classification of digit maps for which, given any periodic point $n$, there are arbitrarily long sequences of consecutive positive integers that end up $n$.

arXiv comment6 pages. Considerably generalized the main result