Abstract
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 fζ that contains a Brjuno rotation f0(z)=z+α, all maps that are conjugate to this rotation form a codimension-1 analytic submanifold near f0.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.
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