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

Abstract

Given a set 𝔸 consisting of positive integers 𝑎 1 < 𝑎 2 < ⋯ < 𝑎 𝑘 and a 𝑘-term partition 𝑃 ∶ 𝑛 1 + 𝑛 2 + ⋯ + 𝑛 𝑘 = 𝑛, find the extremal denominators of the regular and semi-regular continued fraction [0; 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ] with partial quotients 𝑥 𝑖 ∈ 𝔸 and where each 𝑎 𝑖 occurs precisely 𝑛 𝑖 times in the sequence 𝑥 1 , 𝑥 2 , … , 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, an explicit determination of a maximizing arrangement for the semi-regular continuant turned out to be substantially more difficult. Ramharter conjectured that as in the other three cases, the maximizing arrangement is unique (up to reversal) and depends only on the partition 𝑃 and not on the actual values of the 𝑎 𝑖 . He further verified the conjecture in the special case of a binary alphabet. In this paper, we confirm Ramharter's conjecture for sets 𝔸 with |𝔸| = 3 and give an algorithmic procedure for constructing the unique maximizing arrangement. We also show that Ramharter's conjecture fails for sets with |𝔸| ⩾ 4 in that the maximizing

Order Conditions for Languages

“…The order condition can be traced to [15], but its first explicit mention appears in [11] in the particular case of one order and its reverse. Definition 1.…”