HERMAN RINGS AND ARNOLD DISKS

Abstract

For $(\l,a)\in \matbb{C}^*\times \mathbb{C}$, let $f_{\lambda,a}$ be the rational map defined by $f_{\lambda,a}(z) \,{=}\, \lambda z^2 {(az+1)/(z+a)}$. If $\alpha\in \mathbb{R}/\mathbb{Z}$ is a Brjuno number, we let ${\cal D}_\alpha$ be the set of parameters $(\lambda,a)$ such that $f_{\lambda,a}$ has a fixed Herman ring with rotation number $\alpha$ (we consider that $({\it e}^{2i\pi\alpha}{,}0)\,{\in}\, {\cal D}_\alpha$). Results obtained by McMullen and Sullivan imply that, for any $g\in {\cal D}_\alpha$, the connected component of ${\cal D}_\alpha\cap (\mathbb{C}^*\times(\mathbb{C}\setminus \{0,1\}))$ that contains g is isomorphic to a punctured disk.

We show that there is a holomorphic injection $\cal{F}_\alpha\,{:}\,\mathbb{D}\,{\longrightarrow}\, {\cal D}_\alpha$ such that $\cal{F}_\alpha(0) = ({\it e}^{2i\pi \alpha},0)$ and $\cal{F}_\alpha'(0)=(0,r_\alpha),$ where $r_\alpha$ is the conformal radius at 0 of the Siegel disk of the quadratic polynomial $z\longmapsto {\it e}^{2i\pi \alpha}z(1+z)$.

As a consequence, we show that for $a\in (0,1/3)$, if $f_{\l,a}$ has a fixed Herman ring with rotation number $\alpha$ and if $m_a$ is the modulus of the Herman ring, then, as $a\,{\rightarrow}\,0$, we have ${\it e}^{\pi m_a} \,{=} ({r_\alpha}/{a}) + {\cal O}(a).$

We finally explain how to adapt the results to the complex standard family $z\,{\longmapsto} \lambda z {\it e}^{({a}/{2})(z-1/z)}$.