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

TL;DR

This paper explores the self-similar structure of biperiodic Fibonacci words, including special cases like k-Fibonacci and classical Fibonacci words, revealing detailed decomposition patterns that enhance understanding of their recursive nature.

Contribution

It provides a detailed self-similar decomposition of biperiodic Fibonacci words, clarifying the interaction of their recursive components, which was previously only indicated.

Findings

Self-similar decomposition of biperiodic Fibonacci words

Explicit structure of k-Fibonacci and classical Fibonacci words

Enhanced understanding of recursive interactions in these 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.