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

Abstract

We revisit the periodic complexity function h w (n) introduced by Mignosi and Restivo. This function gives the average of the first n local periods of a recurrent infinite word w. We give a different method than that of Mignosi and Restivo for computing the asymptotics of the periodic complexity function of the Thue-Morse word and show how to apply the method to other automatic sequences, like the Rudin-Shapiro word and the period-doubling word.