Sturmian word

In mathematics, a Sturmian word (or billiard sequence), named after Jacques Charles François Sturm, is a certain kind of infinitely long sequence of characters. Such a sequence can be generated by considering a game of English billiards on a square table. The struck ball will successively hit the vertical and horizontal edges, generating a sequence of letters h and v. This sequence is a Sturmian word.

Definition

A word is a (potentially) infinite sequence of symbols drawn from a finite alphabet. Call any finite contiguous subsequence of a word a factor. Then, a word w is Sturmian if, for all natural numbers n, w has exactly n + 1 distinct factors of length n: that is, the complexity function of w is n + 1.

Note that there must then be two distinct factors of length 1, implying that word uses exactly 2 symbols from the alphabet (without loss of generality we can call these 0 and 1). Also, a Sturmian word is necessarily infinite.

Discussion

A sequence $(a_{n})_{n\in \mathbb {N} }$ over {0,1} is a Sturmian word if and only if there exist two real numbers $\alpha$ and $\rho$, with $\alpha$ irrational, such that

$a_{n}=\lfloor \alpha (n+1)+\rho \rfloor -\lfloor \alpha n+\rho \rfloor -\lfloor \alpha \rfloor$

for all $n$. Thus a Sturmian word provides a discretization of the straight line with slope $\alpha$ and intercept ρ. Without loss of generality, we can always assume $0<\alpha <1$, because for any integer k we have

$\lfloor (\alpha +k)(n+1)+\rho \rfloor -\lfloor (\alpha +k)n+\rho \rfloor -\lfloor \alpha +k\rfloor =a_{n}.$

All the Sturmian words corresponding to the same slope $\alpha$ have the same set of factors; the word $c_{\alpha }$ corresponding to the intercept $\rho =0$ is the standard word or characteristic word of slope $\alpha$. Hence, if $0<\alpha <1$, the characteristic word $c_{\alpha }$ is the first difference of the Beatty sequence corresponding to the irrational number $\alpha$.

The standard word $c_{\alpha }$ is also the limit of a sequence of words $(s_{n})_{n\geq 0}$ defined recursively as follows:

Let $[0;d_{1}+1,d_{2},\ldots ,d_{n},\ldots ]$ be the continued fraction expansion of $\alpha$, and define

• $s_{0}=1$
• $s_{1}=0$
• $s_{n+1}=s_{n}^{d_{n}}s_{n-1}{\text{ for }}n>0$

where the product between words is just their concatenation. Every word in the sequence $(s_{n})_{n>0}$ is a prefix of the next ones, so that the sequence itself converges to an infinite word, which is $c_{\alpha }$.

The infinite sequence of words $(s_{n})_{n\geq 0}$ defined by the above recursion is called the standard sequence for the standard word $c_{\alpha }$, and the infinite sequence d = (d₁, d₂, d₃, ...) of nonnegative integers, with d₁ ≥ 0 and d_n > 0 (n ≥ 2), is called its directive sequence.

A famous example of (standard) Sturmian word is the Fibonacci word; its slope is $1/\phi ^{2}$, where $\phi$ is the golden ratio.

History

Although the study of Sturmian words dates back to Johann III Bernoulli (1772), the first comprehensive study of them was by Gustav A. Hedlund and Marston Morse in 1940. They introduced the term Sturmian, in honor of the mathematician Jacques Charles François Sturm.

References

• Lothaire, M. (2002). "Sturmian Words". Algebraic Combinatorics on Words. Cambridge UK: Cambridge University Press. ISBN 0521812208. Retrieved 2007-02-25. {{cite book}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help)

See also

• Cutting sequence