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Deltahedron (talk | contribs) cite Cassaigne & Nicolas (2010) |
Deltahedron (talk | contribs) extended definitions, cite Pytheas Fogg (2002) |
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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 complexity sequence at most ''n''+1.<ref name=L48>Lothaire (2011) p.48</ref>
A [[Sturmian word]] over a binary alphabet is one with complexity function ''n'' + 1.<ref name=PF6>Pytheas Fogg (2002) p.6</ref> A sequence is Sturmian if and only if it is balanced and aperiodic.<ref name=L46>Lothaire (2011) p.46</ref> An example is the [[Fibonacci word]].<ref name=PF6/><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> More generally, a Sturmian word over an alphaber of size ''k'' is one with complexity ''n''+''k''−1. An Arnoux-Rauzy word over a ternary alphabet has complexity 2''n'' + 1:<ref name=PF6/> an example is the [[Tribonacci word]].<ref name=PF368>Pytheas Fogg (2002) p.368</ref>
The ''[[topological entropy]]'' of an infinite sequence ''u'' is defined by
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