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The Markoff Property
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The Markoff property is a combinatorial property of infinite words on the alphabet {a,b}, and of bi-infinite words. Such a word has this property if whenever there is a factor xy in the word,with x,y equal to the letters a,b (in some order), then itmay be extended into a factor of the formym’xymx, wherem’ is the reversal ofm, and where the length ofmis bounded (the bound depends only on the infinite word). As discussed in this chapter, the main theorem, due toMarkoff, is that this property implies periodicity, with a periodic pattern which must be a Christoffel word. It is one of the crucial results inMarkoff’s theory.
Title: The Markoff Property
Description:
The Markoff property is a combinatorial property of infinite words on the alphabet {a,b}, and of bi-infinite words.
Such a word has this property if whenever there is a factor xy in the word,with x,y equal to the letters a,b (in some order), then itmay be extended into a factor of the formym’xymx, wherem’ is the reversal ofm, and where the length ofmis bounded (the bound depends only on the infinite word).
As discussed in this chapter, the main theorem, due toMarkoff, is that this property implies periodicity, with a periodic pattern which must be a Christoffel word.
It is one of the crucial results inMarkoff’s theory.
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