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| [[File:Fibonacci word cutting sequence.png|thumb|350px|The [[Fibonacci word]] is an example of a Sturmian word. The start of the [[cutting sequence]] shown here illustrates the start of the word 0100101001.]]
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| In [[mathematics]], a '''Sturmian word''' ('''Sturmian sequence''' or '''billiard sequence'''<ref>{{cite doi|10.1007/3-540-45535-3_19}}</ref>), named after [[Jacques Charles François Sturm]], is a certain kind of infinitely long [[string (computer science)|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.<ref>{{cite book|page=117|title=Recent Trends in Combinatorics: The Legacy of Paul Erdős|year=2009|first1=Ervin|last1=Győri|first2=Vera|last2=Sós|publisher=Cambridge University Press|isbn=0-521-12004-7}}</ref> This sequence is a Sturmian word.
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| == Definition ==
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| Sturmian sequences can be defined strictly in terms of their combinatoric properties or geometrically as [[cutting sequence]]s for lines of irrational slope or codings for [[irrational rotation]]s. They are traditionally taken to be infinite sequences on the alphabet of the two symbols ''0'' and ''1''.
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| === Combinatoric Definitions ===
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| ==== Sequences of Low Complexity ====
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| For an infinite sequence of symbols ''w'', let ''σ(n)'' be the [[complexity function]] of ''w''; i.e., ''σ(n)'' = the number of distinct subwords in ''w'' of length ''n''. ''w'' is Sturmian if ''σ(n)=n+1'' for all ''n''.
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| ==== Balanced Sequences ====
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| 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 <math>s\in X</math> |''s''|<sub>1</sub>=''k'' or |''s''|<sub>1</sub>=''k''' where |''s''|<sub>1</sub> is the number of ''1''s in ''s''.
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| Let ''w'' be an infinite sequence of ''0''s and ''1''s and let <math>\mathcal L_n(w)</math> denote the set of all length-''n'' subwords of ''w''. The sequence ''w'' is Sturmian if <math>\mathcal L_n(w)</math> 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 ====
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| Let ''w'' be an infinite sequence of ''0''s and ''1''s. The sequence ''w'' is Sturmian if for some <math>x\in[0,1)</math> and some irrational <math>\theta\in(0,\infty)</math>, ''w'' is realized as the [[cutting sequence]] of the line <math>f(t)=\theta t+x</math>.
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| ==== Difference of Beatty Sequences ====
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| Let ''w=(w<sub>n</sub>)'' be an infinite sequence of ''0''s and ''1''s. The sequence ''w'' is Sturmian if for some <math>x\in[0,1)</math> and some irrational <math>\theta\in(0,1)</math>
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| :<math>w_n = \lfloor n\theta + x\rfloor - \lfloor (n-1)\theta + x \rfloor </math>.
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| ==== Coding of Irrational Rotation ====
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| [[File:Sturmian-sequence-from-irrational-rotation.gif|thumb|150px|An animation showing the Sturmian sequence generated by an irrational rotation with ''θ≈0.2882'' and ''x≈0.0789'']]
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| For <math>\theta\in [0,1)</math>, define <math>T_\theta:[0,1)\to[0,1)</math> by <math>t\mapsto t+\theta\mod 1</math>. For <math>x\in[0,1)</math> define the ''θ''-coding of ''x'' to be the sequence ''(x<sub>n</sub>)'' where
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| :<math>x_n=\left\{\begin{array}{cl}1&\text{ if } T_\theta^n(x)\in [0,\theta)\\0&\text{ else}\end{array}\right.</math>.
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| Let ''w'' be an infinite sequence of ''0''s and ''1''s. The sequence ''w'' is Sturmian if for some <math>x\in[0,1)</math> and some irrational <math>\theta\in(0,\infty)</math>, ''w'' is the ''θ''-coding of ''x''.
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| == Discussion ==
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| ===Example===
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| * A famous example of (standard) Sturmian word is the [[Fibonacci word]];<ref name=deluca1995>{{cite journal | journal=Information Processing Letters | year=1995 | pages=307–312 | doi=10.1016/0020-0190(95)00067-M | title=A division property of the Fibonacci word | first=Aldo | last=de Luca | volume=54 | issue=6 }}</ref> its slope is <math>1/\phi</math>, where <math>\phi</math> is the [[golden ratio]].
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| ===Balanced aperiodic sequences===
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| A set ''S'' of finite binary words is ''balanced'' if for each ''n'' the subset ''S''<sub>''n''</sub> of words of length ''n'' has the property that the [[Hamming weight]] of the words in ''S''<sub>''n''</sub> 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''.<ref name=L48>Lothaire (2011) p.48</ref> 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''.<ref name=L22>Lothaire (2011) p.22</ref> A sequence is Sturmian if and only if it is balanced and aperiodic.<ref name=L46>Lothaire (2011) p.46</ref>
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| ===Slope and intercept===
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| A [[sequence]] <math>(a_n)_{n\in\mathbb{N}}</math> over {0,1} is a Sturmian word if and only if there exist two [[real number]]s, the ''slope'' <math>\alpha</math> and the ''intercept'' <math>\rho</math>, with <math>\alpha</math> [[Irrational number|irrational]], such that
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| :<math>a_n=\lfloor\alpha(n+1)+\rho\rfloor -\lfloor\alpha n+\rho\rfloor-\lfloor\alpha\rfloor</math>
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| for all <math>n</math>.<ref name=AS284>Allouche & Shallit (2003) p.284</ref><ref name=PF152>Pytheas Fogg (2002) p.152</ref> Thus a Sturmian word provides a [[discretization]] of the straight line with slope <math>\alpha</math> and intercept ''ρ''. Without loss of generality, we can always assume <math>0<\alpha<1</math>, because for any integer ''k'' we have
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| : <math>\lfloor(\alpha + k)(n + 1) + \rho\rfloor - \lfloor(\alpha+k)n + \rho\rfloor - \lfloor\alpha + k\rfloor = a_n.</math>
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| All the Sturmian words corresponding to the same slope <math>\alpha</math> have the same set of factors; the word <math>c_\alpha</math> corresponding to the intercept <math>\rho=0</math> is the '''standard word''' or '''characteristic word''' of slope <math>\alpha</math>.<ref name=AS283>Allouche & Shallit (2003) p.283</ref> Hence, if <math>0<\alpha<1</math>, the characteristic word <math>c_\alpha</math> is the [[first difference]] of the [[Beatty sequence]] corresponding to the irrational number <math>\alpha</math>.
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| The standard word <math>c_\alpha</math> is also the limit of a sequence of words <math>(s_n)_{n \ge 0}</math> defined recursively as follows:
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| Let <math>[0; d_1+1, d_2, \ldots, d_n, \ldots]</math> be the [[continued fraction]] expansion of <math>\alpha</math>, and define
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| * <math>s_0=1</math>
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| * <math>s_1=0</math>
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| * <math>s_{n+1}=s_n^{d_n}s_{n-1}\text{ for }n>0</math>
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| where the product between words is just their [[concatenation]]. Every word in the sequence <math>(s_n)_{n>0}</math> is a [[Prefix (computer science)|prefix]] of the next ones, so that the sequence itself [[Limit of a sequence|converges]] to an infinite word, which is <math>c_\alpha</math>.
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| The infinite sequence of words <math>(s_n)_{n \ge 0}</math> defined by the above recursion is called the '''standard sequence''' for the standard word <math>c_\alpha</math>, and the infinite sequence ''d'' = (''d''<sub>1</sub>, ''d''<sub>2</sub>, ''d''<sub>3</sub>, ...) of nonnegative integers, with ''d''<sub>1</sub> ≥ 0 and ''d''<sub>''n''</sub> > 0 (''n'' ≥ 2), is called its '''directive sequence'''.
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| A Sturmian word ''w'' over {0,1} is characteristic if and only if both 0''w'' and 1''w'' are Sturmian.<ref name=BS1994>{{citation | last1=Berstel | first1=J. | last2=Séébold | first2=P. | title=A remark on morphic Sturmian words | journal=RAIRO, Inform. Théor. Appl. 2| volume=8 | number=3-4 | pages=255–263 | year=1994 | issn=0988-3754 | zbl=0883.68104 }}</ref>
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| ===Frequencies===
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| If ''s'' is an infinite sequence word and ''w'' is a finite word, let μ<sub>''N''</sub>(''w'') denote the number of occurrences of ''w'' as a factor in the prefix of ''s'' of length ''N''+|''w''|-1. If μ<sub>''N''</sub>(''w'') has a limit as ''N''→∞, we call this the '''frequency''' of ''w'', denoted by μ(''w'').<ref name=LotII83>Lothaire (2011) p.83</ref>
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| For a Sturmian word ''s'', every finite factor has a frequency. The '''three-distance theorem''' states 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.<ref name=LotII83/>
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| ==Non-binary words==
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| 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.<ref name=PF6>Pytheas Fogg (2002) p.6</ref> They can be described in terms of cutting sequences for ''k''-dimensional space.<ref name=PF84>Pytheas Fogg (2002) p.84</ref> An alternative definition is as words of minimal complexity subject to not being ultimately periodic.<ref name=PF85>Pytheas Fogg (2002) p.85</ref>
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| ==Associated real numbers==
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| A real number for which the digits with respect to some fixed base form a Sturmian word is a [[transcendental number]].<ref name=PF85/><ref name=PF64>Pytheas Fogg (2002) p.64</ref>
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| == History ==
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| Although the study of Sturmian words dates back to [[Johann III Bernoulli]] (1772),<ref>J. Bernoulli III, Sur une nouvelle espece de calcul, Recueil pour les Astronomes, vol. 1, Berlin, 1772, pp. 255–284</ref><ref name=AS295>Allouche & Shallit (2003) p.295</ref> it was [[Gustav A. Hedlund]] and [[Marston Morse]] in 1940 who coined the term ''Sturmian'' to refer to such sequences,<ref name=AS295/><ref>{{cite jstor|2371431}}</ref> in honor of the mathematician [[Jacques Charles François Sturm]] due to the relation with the [[Sturm comparison theorem]].<ref name=PF114>Pytheas Fogg (2002) p.114</ref>
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| == See also ==
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| * [[Cutting sequence]]
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| ==References==
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| {{Reflist}}
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| *{{cite book | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}
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| *{{cite book | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-0-521-11169-0 | zbl=pre06066616 }}
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| * {{cite book | last = Lothaire | first = M. | authorlink = M. Lothaire | title = Algebraic Combinatorics on Words | url = http://www-igm.univ-mlv.fr/~berstel/Lothaire/AlgCWContents.html | accessdate = 2007-02-25 | year = 2002 | publisher = [[Cambridge University Press]] | location = Cambridge | isbn = 0-521-81220-8 | chapter = Sturmian Words | zbl=1001.68093 | chapterurl = http://www-igm.univ-mlv.fr/%7Eberstel/Lothaire/ChapitresACW/C2.ps }}
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| * {{cite book | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}
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| [[Category:Combinatorics on words]]
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| [[Category:Sequences and series]]
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I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. I've usually loved living in Alaska. Since he was eighteen he's been operating as an information officer but he plans on altering it. Playing badminton is a factor that he is completely addicted to.
Feel free to surf to my site :: psychic love readings (relevant website trasser.info)