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 @math . A zigzag word is represented by an initial string, followed by a finite list of terms, each of which repeats for each @math in one of three ways: it grows forward [ @math , backward [ @math ], or just occurs once [ @math ]. Each term can recursively contain subterms with their own forward and backward repetitions. We show that an infinite word is morphic with growth @math iff it is a zigzag word of depth @math . As corollaries, we obtain that the morphic words with growth @math are exactly the ultimately periodic words, and the morphic words with growth @math are exactly the multilinear words.
@cite_9: For an alphabet A and a morphism h : A∗ → A∗, the set of words w such that the DOL language L(A, h, w) is a BOUNDED language is shown to be B∗, where B is an effectively computable subset of A. Consequently, BOUNDEDNESS is decidable for DOL languages. The result depends on the authors' recent results on periodic DOL languages. Interpretation of the result for polynomially bounded DOL languages is also considered.
A previous characterization of morphisms with polynomial growth is due to @cite_9 , who introduce the notion of the rank'' of letters under a morphism. They show that a string @math has rank @math under a morphism @math iff @math is the minimal degree of a polynomial @math such that for every @math , @math . We make use of this result in proving the equivalence between morphic words of polynomial growth and zigzag words.