Duality in Biperiodic Fibonacci Words Substitution Frequencies and Combinatorial Invariants

Technical Summary: Duality in Biperiodic Fibonacci Words

Problem Statement
The paper investigates the family of biperiodic Fibonacci words, denoted $F(a,b)$, generated by the directive sequence $(a, b, a, b, \dots)$ with integer parameters $a, b \ge 1$. While the asymptotic letter frequencies of these words depend on a symmetric algebraic quantity $A_a(a,b)$, other invariants, specifically the critical exponent $CE(F(a,b))$, exhibit apparent asymmetry under the exchange of parameters ($a \leftrightarrow b$). The central problem is to explain this asymmetry: is it an artifact of closed-form expressions, or does it reflect a deeper structural relationship between $F(a,b)$ and $F(b,a)$? The paper seeks to determine if an explicit morphism exists that maps $F(b,a)$ to $F(a,b)$ and to analyze how this mapping affects combinatorial invariants such as letter frequencies, return words, and palindromic structures.

Methodology
The authors employ the framework of S-adic systems and Sturmian word theory.

1. S-adic Representation: The paper establishes that $F(a,b)$ coincides with the standard Sturmian sequence generated by the periodic directive sequence $(a, b, a, b, \dots)$. This allows the use of desubstitution identities.
2. Morphism Construction: The core methodological tool is the explicit morphism $\sigma_a: 0 \mapsto 0^a1, 1 \mapsto 0$. The authors utilize induction on the finite approximations $F(a,b)_n$ to prove that $\sigma_a(F(b,a)) = F(a,b)$ exactly, without the need for letter relabeling or bounded prefix corrections.
3. Combinatorial Analysis: Using the established duality $\sigma_a$, the paper derives exact formulas for:
   • Letter frequencies via limit analysis of the morphism's action on block lengths.
   • Return words by analyzing the block decomposition of the infinite word.
   • Palindromic prefixes by leveraging classical results on standard Sturmian sequences and central words.
4. Continued Fractions: The slope $\theta(a,b)$ of the word is analyzed via its continued fraction expansion, linking the combinatorial properties to the quadratic irrational $A(a,b)$.

Key Contributions and Results

• Parity-Shift Duality Theorem: The paper proves that $F(a,b) = \sigma_a(F(b,a))$ for all $a, b \ge 1$. This establishes a precise structural correspondence where the word $F(a,b)$ is the image of $F(b,a)$ under the morphism $\sigma_a$. This explains the asymmetry in invariants as a consequence of the "length-redistribution mechanism" induced by $\sigma_a$.
• Letter Frequencies: The authors derive exact closed-form expressions for the frequencies of letters 0 and 1 in $F(a,b)$:
  $$\text{freq}_1(F(a,b)) = \frac{b}{\alpha + b}, \quad \text{freq}_0(F(a,b)) = \frac{\alpha}{\alpha + b}$$
  where $\alpha = A(a,b)$. This corrects previous assumptions that frequencies might be symmetric under $a \leftrightarrow b$; they are not, unless $a=b$.
• Return Words: The paper provides a complete description of the return words for each letter:
  • The return words for 0 are $\{0, 01\}$, which are independent of $a$ and $b$.
  • The return words for 1 are $\{10^a, 10^{a+1}\}$.
  • The duality acts on the set of return words for 1 by substituting the exponent $a$ with $b$, while the set for 0 remains invariant.
• Sturmian Properties: It is proven that $F(a,b)$ is a standard Sturmian word for all $a, b \ge 1$. Consequently, the balance function is $B(n) \equiv 1$ and the abelian complexity is $AC(n) \equiv 2$ for all $n$. These invariants are trivially symmetric under $a \leftrightarrow b$.
• Palindromic Structure: The paper proves that for every $n \ge 2$, the word obtained by deleting the last two letters of the finite approximation $F(a,b)_n$ is a palindrome. This confirms the existence of arbitrarily long palindromic prefixes.
• Slope and Continued Fraction: The slope $\theta(a,b)$ is determined to have the continued fraction expansion $[0; ab+1, 1, ab]$. The paper demonstrates that the slope and the critical exponent depend on the pair $(a,b)$ only through the product $ab$ and the maximum $\max(a,b)$.
• Critical Exponent Minimization: The paper defines an index $Ind(F(a,b))$ related to the critical exponent and proves that it attains its global minimum uniquely at $(a,b) = (1,1)$, recovering the classical Fibonacci word value $2 + \phi$.

Significance and Claims
The paper claims that the apparent asymmetry in the critical exponent and letter frequencies of biperiodic Fibonacci words is not an isolated phenomenon but a uniform consequence of the structural duality between $F(a,b)$ and $F(b,a)$. By identifying the explicit morphism $\sigma_a$, the authors provide a unified explanation for why invariants dependent on the interaction between letter identity and block length fail to be symmetric under parameter exchange.

The work resolves the "puzzle" of why algebraic quantities like $A(a,b)$ are symmetric while combinatorial invariants are not, attributing the difference to the specific action of the morphism. The paper explicitly states that this duality relationship had not been previously observed. It also identifies open problems, including the computation of the full palindromic complexity function $P_{F(a,b)}(n)$ for all $n$ and the identification of exact extremal repetitions for the critical exponent, noting that current lower bounds are not tight. The authors suggest that the framework could be extended to $k$-periodic directive sequences, implying a broader cyclic duality.