Extremal values of semi-regular continuants and codings of interval exchange transformations
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Given a set A consisting of positive integers a_1<a_2<⋯<a_k and a k-term partition P:n_1+⋯+n_k=n, find the extremal denominators of the regular and semi-regular continued fraction [0;x_1,…,x_n] with partial quotients x_i∈ A, where each a_i occurs precisely n_i times in the sequence x_1,…,x_n. In 1983, G. Ramharter gave an explicit description of the extremal arrangements of the regular continued fraction and the minimizing arrangement for the semi-regular continued fraction and showed that in each case the arrangement is unique up to reversal and independent of the actual values of the positive integers a_i. However, the determination of the maximizing arrangement for the semi-regular continuant turned out to be more difficult. 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 a_i. He further conjectured that this should be true for general A with |A|≥ 2. In this paper we confirm Ramharter's conjecture for sets A with |A|=3 and give an algorithmic procedure for constructing the maximizing arrangement. We also show that Ramharter's conjecture fails in general for |A|≥ 4 in that the maximizing arrangement is neither unique nor independent of the values of the digits in A. The central idea, as discovered by Ramharter, is that the extremal arrangements satisfy a strong combinatorial condition. 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 minima of binary quadratic forms. We show that this same combinatorial condition, 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.
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