Analytic linearization of conformal maps of the annulus.

We consider holomorphic maps defined in an annulus around R / Z in C / Z. E. Risler proved that in a generic analytic family of such maps f ζ that contains a Brjuno rotation f 0 (z) = z + α , 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 R / 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. [ABSTRACT FROM AUTHOR]