[en] The Thue-Morse, Fibonacci-Thue-Morse and Allouche-Johnson words are related to the binary, Zeckendorf and Narayana numeration systems respectively, as they count the parity of the number of ones in representations of natural numbers in those systems. These three numeration systems can be seen as the special cases for k=1,2,3 of a positional numeration system based on the recurrence relation $U_n=U_{n−1}+U_{n−k}$. As such, the three words above can be seen as the first elements in a family of binary words related to those numeration systems.
In a recent preprint, J. Shallit put forth two conjectures on words of this family, the first concerning the first difference of their factor complexity and the second concerning their asymptotic critical exponent. In this talk, we will prove both conjectures by studying bispecial factors within those words. We will highlight a method by K. Klouda allowing us to fully list those bispecial factors and show how this list allows us to attack the conjectures in question. Joint work with L'ubomíra Dvořáková and Edita Pelantová.
