Herman rings and Arnold disks

Abstract

For $(\l,a)\in \C^*\times \C$, let $f_{\l,a}$ be the rational map defined by $$f_{\l,a}(z) = \l z^2 \frac{az+1}{z+a}.$$ If $\a\in \R/\Z$ is a Bruno number, we let ${\cal D}_\a$ be the set of parameters $(\l,a)$ such that $f_{\l,a}$ has a fixed Herman ring with rotation number $\a$ (we consider that $(\ex^{2i\pi\a},0)\in {\cal D}_\a$). The results obtained in \cite{mcs} imply that for any $g\in {\cal D}_\a$ the connected component of ${\cal D}_\a\cap (\C^*\times(\C\setminus \{0,1\}))$ which contains $g$ is isomorphic to a punctured disk. In this article, we show that there is an isomorphism $\F_\a:\D\to {\cal D}_\a$ such that $$\F_\a(0) = (\ex^{2i\pi \a},0)\quad{\rm and}\quad \F_\a'(0)=(0,r_\a),$$ where $r_\a$ is the conformal radius at $0$ of the Siegel disk of the quadratic polynomial $z\mapsto \ex^{2i\pi \a}z(1+z)$. In particular, ${\cal D}_\a$ is a Riemann surface isomorphic to the unit disk. As a consequence, we show that for $a\in (0,1/3)$, if $f_{\l,a}$ has a fixed Herman ring with rotation number $\a$ and if $m_a$ is the modulus of the Herman ring, then, as $a\to 0$, we have \[ \ex^{\pi m_a} = \frac{r_\a}{a} + {\cal O}(a). \] We finally explain how to adapt the results to the complex standard family $z\mapsto \l z \ex^{\frac{a}{2}(z-1/z)}$.