Sturmian word

 ( Combinatorics On Words ) 

In mathematics, a Sturmian word ( Sturmian sequence 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 labelled 0 and 1 generating a sequence of letters.

(2025). 9780521120043, Cambridge University Press. ISBN 9780521120043

This sequence is a Sturmian word.

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Definition Sturmian sequences can be defined strictly in terms of their combinatoric properties or geometrically as for lines of irrational slope or codings for irrational rotations. They are traditionally taken to be infinite sequences on the alphabet of the two symbols 0 and 1.

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Combinatorial definitions

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Sequences of low complexity For an infinite sequence of symbols w, let σ( n) be the complexity function of w; i.e., σ( n) = the number of distinct subword in w of length n. Then w is Sturmian if σ( n) =  n + 1 for all  n.

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Balanced sequences A set X of binary strings is called balanced if the Hamming weight of elements of X takes at most two distinct values. That is, for any $s\backslash in\; X$ | s|₁ =  k or | s|₁ =  k' where | s|₁ is the number of 1s in s.

Let w be an infinite sequence of 0s and 1s and let $\backslash mathcal\; L\_n(w)$ denote the set of all length- n subwords of w. The sequence w is Sturmian if $\backslash mathcal\; L\_n(w)$ is balanced for all n and w is not eventually periodic.

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Geometric definitions

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Cutting sequence of irrational Let w be an infinite sequence of 0s and 1s. The sequence w is Sturmian if for some $x\backslash in[0,1)$ and some irrational $\backslash theta\backslash in(0,\backslash infty)$, w is realized as the cutting sequence of the line $f(t)=\backslash theta\; t+x$.

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Difference of Beatty sequences Let w = ( w _n) be an infinite sequence of 0s and 1s. The sequence w is Sturmian if it is the difference of non-homogeneous , that is, for some $x\backslash in[0,1)$ and some irrational $\backslash theta\backslash in(0,1)$

$w\_n\; =\; \backslash lfloor\; n\backslash theta\; +\; x\backslash rfloor\; -\; \backslash lfloor\; (n-1)\backslash theta\; +\; x\; \backslash rfloor$

for all $n$ or

$w\_n\; =\; \backslash lceil\; n\backslash theta\; +\; x\backslash rceil\; -\; \backslash lceil\; (n-1)\backslash theta\; +\; x\; \backslash rceil$

for all $n$.

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Coding of irrational rotation For $\backslash theta\backslash in\; [0,1)$, define $T\_\backslash theta:[0,1)\backslash to[0,1)$ by $t\backslash mapsto\; t+\backslash theta\backslash bmod\; 1$. For $x\backslash in[0,1)$ define the θ-coding of x to be the sequence ( x _n) where

Let w be an infinite sequence of 0s and 1s. The sequence w is Sturmian if for some $x\backslash in[0,1)$ and some irrational $\backslash theta\backslash in(0,\backslash infty)$, w is the θ-coding of  x.

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Discussion

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Example A famous example of a (standard) Sturmian word is the Fibonacci word; its slope is $1/\backslash phi$, where $\backslash phi$ is the golden ratio.

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Balanced aperiodic sequences A set S of finite binary words is balanced if for each n the subset S _n of words of length n has the property that the Hamming weight of the words in S _n takes at most two distinct values. A balanced sequence is one for which the set of factors is balanced. A balanced sequence has at most n+1 distinct factors of length n.

M. Lothaire (2025). 9780521812207, Cambridge University Press. . ISBN 9780521812207

An aperiodic sequence is one which does not consist of a finite sequence followed by a finite cycle. An aperiodic sequence has at least n + 1 distinct factors of length n. A sequence is Sturmian if and only if it is balanced and aperiodic.

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Slope and intercept

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

$a\_n=\backslash lfloor\backslash alpha(n+1)+\backslash rho\backslash rfloor\; -\backslash lfloor\backslash alpha\; n+\backslash rho\backslash rfloor-\backslash lfloor\backslash alpha\backslash rfloor$

for all $n$.

(2025). 9780521823326, Cambridge University Press. ISBN 9780521823326

N. Pytheas Fogg (2025). 9783540441410, Springer-Verlag. ISBN 9783540441410

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

$\backslash lfloor(\backslash alpha\; +\; k)(n\; +\; 1)\; +\; \backslash rho\backslash rfloor\; -\; \backslash lfloor(\backslash alpha+k)n\; +\; \backslash rho\backslash rfloor\; -\; \backslash lfloor\backslash alpha\; +\; k\backslash rfloor\; =\; a\_n.$

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

The standard word $c\_\backslash alpha$ is also the limit of a sequence of words $(s\_n)\_\{n\; \backslash ge\; 0\}$ defined recursively as follows:

Let $0;$ be the continued fraction expansion of $\backslash alpha$, and define

• $s\_0=1$
• $s\_1=0$
• $s\_\{n+1\}=s\_n^\{d\_n\}s\_\{n-1\}\backslash text\{\; for\; \}n>0$

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 $c\_\backslash alpha$.

The infinite sequence of words $(s\_n)\_\{n\; \backslash ge\; 0\}$ defined by the above recursion is called the standard sequence for the standard word $c\_\backslash alpha$, 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 Sturmian word w over {0,1} is characteristic if and only if both 0 w and 1 w are Sturmian.

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Frequencies

If s is an infinite sequence word and w is a finite word, let μ _N( w) denote the number of occurrences of w as a factor in the prefix of s of length N + | w| − 1. If μ _N( w) has a limit as N→∞, we call this the frequency of w, denoted by μ( w).

For a Sturmian word s, every finite factor has a frequency. The three-gap theorem implies that the factors of fixed length n have at most three distinct frequencies, and if there are three values then one is the sum of the other two.

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Non-binary words

For words over an alphabet of size k greater than 2, we define a Sturmian word to be one with complexity function n +  k − 1. They can be described in terms of cutting sequences for k-dimensional space. An alternative definition is as words of minimal complexity subject to not being ultimately periodic.

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Associated real numbers

A real number for which the digits with respect to some fixed base form a Sturmian word is a transcendental number.

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Sturmian endomorphisms

An endomorphism of the free monoid B^∗ on a 2-letter alphabet B is Sturmian if it maps every Sturmian word to a Sturmian word and locally Sturmian if it maps some Sturmian word to a Sturmian word. The Sturmian endomorphisms form a submonoid of the monoid of endomorphisms of B^∗.

Define endomorphisms φ and ψ of B^∗, where B = {0,1}, by φ(0) = 01, φ(1) = 0 and ψ(0) = 10, ψ(1) = 0. Then I, φ and ψ are Sturmian, and the Sturmian endomorphisms of B^∗ are precisely those endomorphisms in the submonoid of the endomorphism monoid generated by { I,φ,ψ}.

A morphism is Sturmian if and only if the image of the word 10010010100101 is a balanced sequence; that is, for each n, the of the subwords of length n take at most two distinct values.

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History

Although the study of Sturmian words dates back to Johann III Bernoulli (1772),J. Bernoulli III, Sur une nouvelle espece de calcul, Recueil pour les Astronomes, vol. 1, Berlin, 1772, pp. 255–284 it was Gustav A. Hedlund and Marston Morse in 1940 who coined the term Sturmian to refer to such sequences, in honor of the mathematician Jacques Charles François Sturm due to the relation with the Sturm comparison theorem.

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See also

• Cutting sequence
• Word (group theory)
• Morphic word
• Lyndon word

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Further reading

• Yann Bugeaud (2025). 9780521111690, Cambridge University Press. ISBN 9780521111690
• M. Lothaire (2025). 9780521180719, Cambridge University Press. ISBN 9780521180719

Categories: Combinatorics On Words, Sequences And Series

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