Extremal values of semi‐regular continuants and codings of interval exchange transformations

Given a set A $\mathbb {A}$ consisting of positive integers 1 < 2 ⋯ k $a_1<a_2<\cdots <a_k$ and k-term partition P : n + = $P: n_1+n_2 \cdots n_k=n$ , find the extremal denominators regular semi-regular continued fraction [ 0 ; x … ] $[0;x_1,x_2,\ldots ,x_n]$ with partial quotients i ∈ $x_i\in \mathbb where each $a_i$ occurs precisely $n_i$ times in sequence $x_1,x_2,\ldots ,x_n$ . In 1983, G. Ramharter gave an explicit description arrangements minimizing arrangement for showed that case is unique up to reversal independent actual values However, determination maximizing continuant turned out be substantially more difficult. conjectured as other three cases, (up reversal) depends only on not He further verified conjecture special binary alphabet. this paper, we confirm Ramharter's sets | 3 $|\mathbb {A}|=3$ give algorithmic procedure constructing arrangement. We also show fails ⩾ 4 {A}|\geqslant 4$ general neither nor digits The central idea satisfy strong combinatorial condition. This condition may stated or less verbatum context infinite sequences ordered set. bi-infinite words, coincides Markoff property, discovered by A. 1879 his study minima quadratic forms. same fundamental property which describes orbit structure natural codings points under symmetric k-interval exchange transformation.