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

TL;DR

This paper introduces a new method for analyzing the asymptotic behavior of the periodic complexity function in certain automatic sequences, including Thue-Morse, Rudin-Shapiro, and period-doubling words.

Contribution

It presents an alternative approach to compute the asymptotics of the periodic complexity function for these sequences, expanding the analytical tools available.

Findings

New method for asymptotic analysis of periodic complexity functions

Application of the method to Thue-Morse, Rudin-Shapiro, and period-doubling sequences

Enhanced understanding of the local period structure in automatic sequences

Abstract

We revisit the periodic complexity function $h_{\bf w}(n)$ introduced by Mignosi and Restivo. This function gives the average of the first $n$ local periods of a recurrent infinite word ${\bf w}$. 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.