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In this paper we study the privileged complexity function of the Thue-Morseword. We prove a recursive formula describing this function, and using theformula we show that the function is unbounded and that the values of thefunction have arbitrarily large gaps of zeros. This demonstrates that theprivileged complexity function of an infinite word can drastically differ fromits palindromic complexity function, even though there are relations betweenthese functions. Further we study the behavior of palindromes and privilegedwords in infinite words and the relation between rich words and privilegedwords.

Privileged Factors in the Thue-Morse Word - A Comparison of Privileged Words and Palindromes