On the Iterates of Digit Maps

TL;DR

This paper classifies digit maps based on their iterative behavior, showing that for any periodic point, there exist arbitrarily long sequences of consecutive integers converging to it under iteration.

Contribution

It provides a complete classification of digit maps regarding the existence of arbitrarily long sequences converging to any given periodic point.

Findings

Classified digit maps with respect to their iterative convergence properties.

Extended previous results from specific cases like base 10 and quadratic functions.

Established conditions for the existence of long sequences ending at any periodic point.

Abstract

Given a base $b$, a "digit map" is a map $f: \mathbb{Z}^{\ge 0} \to \mathbb{Z}^{\ge 0}$ of the form $f(\sum_{i=0}^n a_ib^i) = \sum_{i=0}^n f_*(a_i)$, $0 \le a_i \le b-1$ for each $i$, where $f_* : \{0,1,\dots, b-1\} \to \mathbb{Z}^{\ge 0}$ satisfies $f_*(0) = 0$ and $f_*(1) = 1$. It has been proven for $b=10$ and $f_*(m) = m^2$, 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 $n$.