 
Summary: REVERSALS AND PALINDROMES IN CONTINUED
FRACTIONS
BORIS ADAMCZEWSKI AND JEANPAUL ALLOUCHE
Abstract. Several results on continued fractions expansions are direct
on indirect consequences of the mirror formula. We survey occurrences
of this formula for Sturmian real numbers, for (simultaneous) Diophan
tine approximation, and for formal power series.
1. Introduction
In the present survey, a conference version of which appeared as [1], we
will focus on reversals of patterns and on palindromic patterns that occur
in continued fraction expansions for real numbers and for formal Laurent
series with coefficients in a finite field. Our main motivation comes from the
remark that various very recent, and apparently unrelated, works make use
of an elementary formula for continued fractions, referred to as the mirror
formula all along this paper (see for example [3, 4, 6, 5, 7, 15, 19, 21, 22,
44, 75, 74] for related papers published since 2005). This leads us to review
some of these results, together with older ones, and to underline the central
r^ole played by this formula.
The first part of the paper (Sections 4 and 5) deals with combinatorics
on words. We investigate in particular some questions related to the criti
