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A Characterization of Morphic Words with Polynomial Growth
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 .
A zigzag word is represented by an initial string, followed by a finite list of
terms, each of which repeats for each in one of three ways: it grows
forward [, backward [], or
just occurs once []. Each term can recursively contain subterms with their
own forward and backward repetitions. We show that an infinite word is morphic
with growth iff it is a zigzag word of depth . As corollaries,
we obtain that the morphic words with growth are exactly the ultimately
periodic words, and the morphic words with growth are exactly the
multilinear words