# A Characterization of Morphic Words with Polynomial Growth

## TL;DR

This paper characterizes morphic words with polynomial growth using a new concept called zigzag words, linking their growth rate to the depth of the zigzag structure, and classifies words with linear and quadratic growth.

## Contribution

It introduces zigzag words as a new framework to characterize morphic words with polynomial growth and establishes a precise correspondence between growth rate and zigzag depth.

## Key findings

- Morphic words with growth Θ(n^k) are exactly zigzag words of depth k.
- Morphic words with linear growth are exactly ultimately periodic words.
- Morphic words with quadratic growth are exactly multilinear words.

## Abstract

A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a $\textit{zigzag word}$. A zigzag word is represented by an initial string, followed by a finite list of terms, each of which repeats for each $n \geq 1$ in one of three ways: it grows forward [$t(1)\ t(2)\ \dotsm\ t(n)]$, backward [$t(n)\ \dotsm\ t(2)\ t(1)$], or just occurs once [$t$]. Each term can recursively contain subterms with their own forward and backward repetitions. We show that an infinite word is morphic with growth $\Theta(n^k)$ iff it is a zigzag word of depth $k$. As corollaries, we obtain that the morphic words with growth $O(n)$ are exactly the ultimately periodic words, and the morphic words with growth $O(n^2)$ are exactly the multilinear words.