Sturmian Characteristic Word

• Sturmian characteristic word is an infinite binary sequence defined by an irrational slope and its continued fraction expansion, yielding exactly n+1 distinct factors of length n.
• It is generated by standard morphisms prescribed by a directive sequence, resulting in a fixed point that is balanced and uniformly recurrent.
• The detailed combinatorial decomposition into singular and adjoining words interconnects symbolic dynamics, number theory, and quasicrystal theory for algorithmic analysis.

A Sturmian characteristic word is an infinite binary word $c_\alpha$ associated to an irrational slope $\alpha\in(0,1)$; it arises as the canonical symbolic coding of the minimal aperiodic sequences with factor complexity $n+1$ for each length $n$. The combinatorics, structure, and generation of $c_\alpha$ are governed by the continued fraction expansion of $\alpha$, encoded in a directive sequence that prescribes standard morphisms whose fixed points are precisely the characteristic words. The theory connects word combinatorics, symbolic dynamics, continued fractions, morphic substitutions, and number theory in a unified framework that supports explicit decomposition results, factorization, algorithmic generation, and deep classifications of infinite aperiodic order.

1. Definition, Standard Sequences, and Minimal Complexity

A characteristic Sturmian word $c_\alpha$ over an alphabet $\mathcal{A} = \{a, b\}$ is determined for irrational $\alpha \in (0,1)$ by its continued fraction

$$\alpha = [0; 1 + d_1, d_2, \dots], \quad d_i > 0, \ d_1 \geq 1.$$

Given the directive sequence $(d_1, d_2, \ldots)$, define the standard sequence recursively: $$s_{-1} = b, \quad s_0 = a, \quad s_n = s_{n-1}^{d_n} s_{n-2} \quad (n \geq 1).$$ The infinite word

$c_\alpha = \lim_{n \to \infty} s_n$

is the characteristic Sturmian word of slope $\alpha$. It satisfies

$$\forall n \geq 0, \quad \mathrm{Card}(\{\text{factors of length }n\text{ in }c_\alpha\}) = n + 1,$$

giving the minimal complexity for an aperiodic word. $c_{1-\alpha}$ is defined analogously as the characteristic word for slope $1-\alpha$.

These words are balanced (any two factors of equal length differ by at most one in the number of each letter), uniformly recurrent, and serve as the canonical "quasicrystals" in symbolic combinatorics (0708.4387).

2. Generation via Morphisms and Fixed Point Structure

Sturmian characteristic words are generated by "standard" morphisms built from the directive sequence. Consider the morphism $\sigma$ associated with the continued fraction expansion as in (0708.4387):

• The standard morphism $\sigma$ is defined so that for all $m$, $|\sigma^m(a b)| = |s_{m+1}|$.
• The letter exchange $E$ (involution $a\leftrightarrow b$) defines $\hat{\sigma} = E \sigma E$.

Fixed point property: $c_\alpha$ is a fixed point of any power of the standard morphism: $$c_\alpha = \lim_{m\rightarrow \infty} \sigma^m(a), \qquad c_{1-\alpha} = \lim_{m\rightarrow \infty} \hat{\sigma}^m(b),$$ if and only if $\alpha = [0; 1 + d_1, d_2, \ldots, d_n]$ with $d_n \geq d_1 \geq 1$.

Thus, the morphic generation process is entirely determined by the continued fraction of $\alpha$, and the standard sequence reflects the combinatorial structure of $c_\alpha$ mirrored in the inflation structure of $\sigma$ (0708.4387).

3. Conjugates, Singular Decomposition, and Structure

Conjugation extends classically: for infinite $x$ and $k \in \mathbb{N}$, the $k$-th conjugate is the infinite word with prefix of length $k$ removed. The central result of (0708.4387) is that every conjugate of $c_\alpha$ admits a decomposition into "generalized adjoining singular words" (Melançon's singular word decomposition).

For $k = q_{m+1} - p$, $2 \leq p \leq q_{m+1} - q_m + 1$, where $(q_n)$ are denominators of convergents of $\alpha$, the $k$-th conjugate

$(\sigma^m)^k(c_\alpha)$

admits a decomposition

$$(\sigma^m)^k(c_\alpha) = u^{-1}(U_m U_{m+1} U_{m+2}\ldots),$$

where $U_j$ are (generalized) adjoining singular words determined from the standard sequence $s_j$ and $u$ is a prefix of a word $V_{m-1}$ also expressible via $q_m$, $p$. This generalizes decompositions previously available for the Fibonacci word ($\alpha = [0;2,1]$).

The original singular word decomposition of $c_\alpha$ takes the form

$c_\alpha = W_{-1} U_0 U_1 U_2 \ldots$

with singular $W_n$ and adjoining singular $U_n$ built in terms of $s_n$ (0708.4387).

4. Continued Fraction Expansion and Recurrence Structure

The continued fraction expansion is fundamental:

• The directive sequence $(d_1, d_2, \ldots)$ both determines the standard sequence, and prescribes the combinatorial inflation for $c_\alpha$ via $\sigma$.
• The lengths of $s_n$ obey $|s_n| = q_n$, where $q_n$ is the denominator of the $n$-th convergent to $\alpha$: $$q_0 = 1, \quad q_1 = 1 + d_1, \quad q_n = d_n q_{n-1} + q_{n-2}.$$
• The decomposition formulae for conjugates of $c_\alpha$ involve arithmetic relationships directly in terms of $q_{n}$ and the partial quotients $d_n$.

Thus, continued fraction expansions of slopes are not just auxiliary number-theoretic data—they impose the full combinatorial "directive order" on $c_\alpha$ and its conjugates (0708.4387).

5. Explicit Example: the Case $\alpha = [0;2,7]$

When $\alpha = [0;2, r]$ (e.g., $r = 7$), (0708.4387) gives the complete singular decomposition for every conjugate:

• The standard morphism $\sigma$ has $|\sigma^m(a b)| = |s_{m+1}| = q_{m+1}$ for every $m$.
• For each $m$, there exists a word $V_m$ of length $|V_m| = q_{m+2} - q_{m+1}$ involved in the singular decomposition of conjugates.
• For $k = q_{m+1} - p$ (with $2 \leq p \leq q_{m+1}-q_m+1$), one has

$(\sigma^m)^k(c_\alpha) = u^{-1}(U_j)_{j\geq m},$

where $u$ is a prefix of $V_{m-1}$, and the decomposition is fully explicit in terms of the standard sequence and continued fraction data.

This generalizes the earlier results for the infinite Fibonacci word and highlights the explicit influence of partial quotients and convergents on the combinatorics of conjugate decompositions (0708.4387).

6. Applications and Broader Context

The decompositions and structure furnished for characteristic Sturmian words generated by morphisms have several important ramifications:

• In symbolic dynamics and combinatorics on words, these decompositions are essential for analyzing recurrence, palindromic factors, and fine balance properties within minimal complexity infinite words.
• In number theory, the link between continued fractions and Sturmian word generation produces a direct bridge to Diophantine approximation and cutting-sequence algorithms.
• In theoretical computer science, especially pattern recognition and formal languages, singular and adjoining singular word factorizations support efficient string matching and the analysis of self-similarity in aperiodic sequences.
• In quasicrystal theory and aperiodic physical models, the self-similarity and hierarchical structure elucidated by morphic decompositions model hierarchical order and tiling properties.

The explicit decomposition results—where each conjugate can be built from generalized adjoining singular words whose structure is tightly controlled by the continued fraction coefficients—equip practitioners with robust combinatorial and algorithmic tools for dissecting and analyzing Sturmian characteristic words in a variety of mathematical and applied settings (0708.4387).