Markov word patterns and a relation on α-words

We prove that all suffixes of a two-way infinite extension of an irrational characteristic word can be represented as Markov word patterns (MWPs) of type 1 and under some conditions, as MWPs of type 2. For each of these MWPs, the pairs of seed words are identified and are described in several different ways. We prove that the simple Sturmian words are the only Sturmian words that are representable as MWPs, each pair of seed words of which is either a pair of adjacent α-words or adjacent α-words with reverse order.