We consider the space $D_\epsilon$ of holomorphic maps defined in annuli around $\mathbb{R}/\mathbb{Z}$. The maps do not necessarily preserve the real axis.
E. Risler proved the following theorem: in any finite-parameter analytic family of such maps with $f_0=R_\phi$, the condition ``$f$ is analytically conjugate to the real rotation $R_\phi$'' is a codimension-1 analytic condition in a neighborhood of $R_\phi$ provided that the angle of rotation $\phi$ is a Bruno number.
We obtain an easy proof of Risler's theorem. We define a (compact analytic) renormalization operator $R$ in $D_\epsilon$ and prove its hyperbolicity with one (complex) unstable direction. The construction of $R$ is inspired both by Yoccoz's renormalizations for circle diffeomorphisms and by the cylinder renormalization. Risler's theorem is a corollary of the existence and analyticity of the stable foliation of $R$.
Joint work with M. Yampolsky.