The LempelZiv Complexity of Fixed Points of Morphisms

The LempelZiv 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.