Analytic linearization of conformal maps of the annulus

TL;DR

This paper introduces a renormalization operator for holomorphic maps on an annulus, demonstrating its hyperbolic structure and characterizing conjugacy to rotations, thereby providing a new proof of Risler's result.

Contribution

It constructs a hyperbolic renormalization operator on univalent maps, offering a novel approach to understanding conjugacy to rotations in annular holomorphic maps.

Findings

The renormalization operator is hyperbolic with a single unstable direction.

Stable foliation consists of maps conjugate to rotations.

Provides a new proof of Risler's theorem using holomorphic motions and Yoccoz's theorem.

Abstract

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.