  We consider holomorphic maps defined in an annulus around $\mathbb R/\mathbb
Z$ in $\mathbb C/\mathbb Z$. E. Risler proved that in a generic analytic family
of such maps $f_\zeta$ that contains a Brjuno rotation $f_0(z)=z+\alpha$, all
maps that are conjugate to this rotation form a codimension-1 analytic
submanifold near $f_0$.
  In this paper, we obtain the Risler's result as a corollary of the following
construction. We introduce a renormalization operator on the space of univalent
maps in a neighborhood of $\mathbb R/\mathbb Z$. We prove that this operator is
hyperbolic, with one unstable direction corresponding to translations. We
further use a holomorphic motions argument and Yoccoz's theorem to show that
its stable foliation consists of diffeomorphisms that are conjugate to
rotations.
