Sturmian word

Sturmian word

In mathematics, a Sturmian word, named after Jacques Charles François Sturm, is a certain kind of infinite word.

Contents

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 α and ρ, with α 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 α and intercept ρ. Without loss of generality, we can always assume 0 < α < 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 α have the same set of factors; the word cα corresponding to the intercept ρ = 0 is the standard word or characteristic word of slope α. Hence, if 0 < α < 1, the characteristic word cα is the first difference of the Beatty sequence corresponding to the irrational number α.

The standard word cα is also the limit of a sequence of words (s_n)_{n \ge 0} defined recursively as follows:

Let [0; d_1+1, d_2, \ldots, d_n, \ldots] be the continued fraction expansion of α, and define

  • s0 = 1
  • s1 = 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 (sn)n > 0 is a prefix of the next ones, so that the sequence itself converges to an infinite word, which is cα.

The infinite sequence of words (s_n)_{n \ge 0} defined by the above recursion is called the standard sequence for the standard word cα, and the infinite sequence d = (d1, d2, d3, ...) of nonnegative integers, with d1 ≥ 0 and dn > 0 (n ≥ 2), is called its directive sequence.

A famous example of (standard) Sturmian word is the Fibonacci word;[1] its slope is 1 / ϕ2, where ϕ 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

  1. ^ de Luca, Aldo (1995). "A division property of the Fibonacci word". Information Processing Letters 54 (6): 307–312. doi:10.1016/0020-0190(95)00067-M. 

See also


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