In this paper, we focus on a new type of complexity of infinite words called privileged complexity. We apply this concept to the ternary Thue-Morse word. First, we present properties of the return words in the ternary Thue-Morse word. Then, we study the privileged words of this infinite word. Finally, we derive a recursive formula that allows us to determine the privileged complexity function of this ternary word.

Infinite words, Frist complete return word, Privileged word, Privileged complexity