On the distribution function of the complexity of finite sequences
Information Sciences: an International Journal
Repetitions in strings: Algorithms and combinatorics
Theoretical Computer Science
Nonlinear dimensionality reduction by locally linear inlaying
IEEE Transactions on Neural Networks
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The Lempel-Ziv complexity is a fundamental measure of complexity for words, closely connected with the famous LZ77 compression algorithm. We investigate this complexity measure for one of the most important families of infinite words in combinatorics, namely, the fixed points of morphisms. We give a complete characterization of the complexity classes which are $\Theta(1)$, $\Theta(\log n)$, and $\Theta(n^{1/k})$, $k \in \mathbb{N}$, $k\ge 2$, depending on the periodicity of the word and the growth function of the morphism. The relation with the well-known classification of Ehrenfeucht, Lee, Rozenberg, and Pansiot for factor complexity classes is also investigated. The two measures complete each other, giving an improved picture for the complexity of these infinite words.