  We consider X 1 ,. .. , X n a sample of data on the circle S 1 , whose
distribution is a twocomponent mixture. Denoting R and Q two rotations on S 1 ,
the density of the X i 's is assumed to be g(x) = pf (R --1 x) + (1 -- p)f (Q
--1 x), where p $\in$ (0, 1) and f is an unknown density on the circle. In this
paper we estimate both the parametric part $\theta$ = (p, R, Q) and the
nonparametric part f. The specific problems of identifiability on the circle
are studied. A consistent estimator of $\theta$ is introduced and its
asymptotic normality is proved. We propose a Fourier-based estimator of f with
a penalized criterion to choose the resolution level. We show that our adaptive
estimator is optimal from the oracle and minimax points of view when the
density belongs to a Sobolev ball. Our method is illustrated by numerical
simulations.
