Geodesic
A Characterization of Morphic Words with Polynomial Growth
Discrete mathematics & theoretical computer science, Tome 22 (2020-2021) no. 1.

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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.
DOI : 10.23638/DMTCS-22-1-3
Classification : 68R15 
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Smith, Tim. A Characterization of Morphic Words with Polynomial Growth. Discrete mathematics & theoretical computer science, Tome 22 (2020-2021) no. 1. doi : 10.23638/DMTCS-22-1-3. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-22-1-3/

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