  Let $f$ be a $C^{1+bv}$ circle diffeomorphism with irrational rotation
number. As established by Douady and Yoccoz in the eighties, for any given
$s>0$ there exists a unique automorphic measure of exponent $s$ for $f$. In the
present paper we prove that the same holds for multicritical circle maps, and
we provide two applications of this result. The first one, is to prove that the
space of invariant distributions of order 1 of any given multicritical circle
map is one-dimensional, spanned by the unique invariant measure. The second
one, is an improvement over the Denjoy-Koksma inequality for multicritical
circle maps and absolutely continuous observables.
