The Lempel-Ziv complexity is a fundamental measure of complexity for words, closely connected with the famous LZ77, LZ78 compression algorithms. 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 characterisation of the complexity classes which are Θ(1), Θ(log n), and Θ(n~(1/k)), k ∈ N, k ≥ 2, depending on the periodicity of the word and the growth function of the morphism. The relation with the well-known classification of Ehren-feucht, 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.
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