Duality in Biperiodic Fibonacci Words Substitution Frequencies and Combinatorial Invariants

Abstract

In this paper, a natural duality on the family of biperiodic Fibonacci words ${\mathfrak{F}}^{(a,b)}$ generated by the directive sequence $(a,b,a,b,\dots )$. Both of ${\mathfrak{F}}^{(a,b)}$ and ${\mathfrak{F}}^{(b,a)}$ are related by the explicit morphism ${\sigma }_a:0\mapsto 0^{a}1$, $1\mapsto 0$, establishing a precise substitutional correspondence between them. We compute the exact letter frequencies, give a complete description of the return words for each letter, prove the existence of arbitrarily long palindromic prefixes, and determine the continued fraction expansion of the slope ${\theta }^{(a,b)}$. These findings reveal that the apparent asymmetry in several invariants arises uniformly from the length-redistribution mechanism induced by the morphism ${\sigma }_a$.

16 pages