Self-Similar Structure of $k$- and Biperiodic Fibonacci Words

Abstract

Defining the biperiodic Fibonacci words as a class of words over the alphabet $\{0,1\}$, and two specializations the $k-$Fibonacci and classical Fibonacci words, we provide a self-similar decomposition of these words into overlapping words of the same type. These self-similar decompositions complement the previous literature where self-similarity was indicated but the specific structure of how the pieces interact was left undiscussed.