In 1983, G. Ramharter gave an explicit description of the extremal arrangements of the regular continued fraction and theminimizing arrangement for the semi-regular continued fraction . He showed that if $|A|=2$ then the maximizing arrangement is unique(up to reversal) and depends only on the partition $P$ and not on the values of the digits in $A$ . He further conjectured that this should be true for general $A$. In this paper we give an algorithmic procedure for constructing the maximizingarrangement . We also show that the same combinatorialcondition, in the framework of infinite words over a $k$-letter alphabet, is the characterizing property which describes the orbit structure of codings of points under a symmetric $k-interval exchange transformation . In the context of bi-infinite binary words, this condition coincides with the Markoff property, discovered by A.A. Markoff in 1879 in his study of binary quadratic forms, we show that this property is the property . We show that it is the same property, which is characterizingproperty, is also characterizing the . orbit structure that describes the orbits of the codings . ofcodings ofpoints under a . symmetric$k $k

Author(s) : Alessandro De Luca, Marcia Edson, Luca Q. Zamboni

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Keywords : property - regular - codings - show - semi -

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