Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extremal values of semi-regular continuants and codings of interval
  exchange transformations   Given a set $A$ consisting of positive integers $a_1<a_2<\cdots<a_k$ and a
$k$-term partition $P:n_1+\cdots+n_k=n$, find the extremal denominators of the
regular and semi-regular continued fraction $[0;x_1,\ldots,x_n]$ with partial
quotients $x_i\in A$, where each $a_i$ occurs precisely $n_i$ times in the
sequence $x_1,\ldots,x_n$. 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
$|A|\geq 2.$ 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
$|A|\geq 4$ 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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The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: comp-gas
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The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-ex