Permutation Complexity and the Letter Doubling Map

Given a countable set X (usually taken to be N or Z), an infinite permutation $\pi$ of X is a linear ordering ${<}_{\pi }$ of X. This paper investigates the combinatorial complexity of infinite permutations on N associated with the image of uniformly recurrent aperiodic binary words under the letter doubling map. An upper bound for the complexity is found for general words, and a formula for the complexity is established for the Sturmian words and the Thue-Morse word.