Maximal pattern complexity of words over letters

The maximal pattern complexity function p α ∗ ( k ) of an infinite word α = α 0 α 1 α 2 ⋯ over ℓ letters, is introduced and studied by [3,4]. present paper we introduce two new techniques, the ascending chain of alphabets and the singular decomposition, to study the maximal pattern complexity. It is shown that if p α ∗ ( k ) < ℓ k holds for some k ≥ 1 , then α is periodic by projection. Accordingly we define a pattern Sturmian word over ℓ letters to be a word which is not periodic by projection and has maximal pattern complexity function p α ∗ ( k ) = ℓ k . Two classes of pattern Sturmian words are given. This generalizes the definition and results of [3] where ℓ = 2 .