Analytic linearization of conformal maps of the annulus

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 $\mathbb{R}\mathrm{/}\mathbb{Z}$. 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.