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We revisit the periodic complexity function $h_{\bf w}(n)$ introduced byMignosi and Restivo. This function gives the average of the first $n$ localperiods of a recurrent infinite word ${\bf w}$. We give a different method thanthat of Mignosi and Restivo for computing the asymptotics of the periodiccomplexity function of the Thue-Morse word and show how to apply the method toother automatic sequences, like the Rudin-Shapiro word and the period-doublingword.

The periodic complexity function of the Thue-Morse word, the Rudin-Shapiro word, and the period-doubling word