Distortion control of conjugacies between quadratic polynomials
The paper is devoted to a theorem on the dynamics of the quadratic polynomial family \(Q_c(z):=z^2+c\) with \(c\in\mathbb C\). For each \(c\), \[ K_c:=\{z\in\mathbb C:(Q_c^{\circ n}(z))_{n\in\mathbb N}\text{ is bounded}\} \] is a compact set called the filled-in Julia set. It is known that this set is connected if and only if \(c\in K_c\), and the set of all such \(c\) is called the Mandelbrot set, which we denote by \(\mathcal M\). Let \(c\in\mathcal M\). To study the boundary of \(K_c\) one employs the conformal (onto) mapping \(\phi_c:\overline{\mathbb C}\setminus K_c\to\{z\in\overline{\mathbb C}:|z|>1\}\) normalized by \(\phi_c(\infty)=\infty\) and \(\phi_c'(\infty)>0\). Given \(\theta\in\mathbb R\), the set \[ R_c(\theta):=\{\phi_c^{-1}(re^{2\pi i\theta}):r>1\} \] is called the dynamic \(\theta\)-ray for \(Q_c\). In a similar way one defines parameter rays. Indeed, it is known that complement \(\overline{\mathbb C}\setminus \mathcal M\) of the Mandelbrot set is a simply connected domain containing~\(\infty\). Let \(\Phi\) be the conformal mapping of this domain onto \(\{z\in\overline{\mathbb C}:|z|>1\}\) normalized by \(\Phi(\infty)=\infty\) and \(\Phi'(\infty)>0\). For \(\theta\in\mathbb R\), the parameter \(\theta\)-ray is defined as the set \[ R_M(\theta):=\{\Phi^{-1}(re^{2\pi i\theta}):r>1\}. \] We say that the dynamic ray \(R_c(\theta)\) (or parameter ray \(R_M\)) lands at a point \(p\) if \(\phi_c^{-1}(re^{2\pi i\theta})\to p\) (or \(\Phi^{-1}(re^{2\pi i\theta})\to p\), respectively) as \(r\to1+0\). The image \(Q_c(R_c(\theta))\) of any dynamic ray \(R_c(\theta)\) is again a dynamic ray, namely~\(R_c(2\theta)\). A dynamic ray~\(R_c(\theta)\) is called periodic if \(Q^{\circ n}_c(R_c(\theta))=R_c(\theta)\) for some \(n\in\mathbb N\), and strictly preperiodic if it is not periodic but \(Q^{\circ m}_c(R_c(\theta))\) is periodic for some \(m\in\mathbb N\). (Clearly, this depends only on \(\theta\), but not on~\(c\).) The following result is due to Douady and Hubbard. \textbf{Theorem 1.} Let \(c\in\mathcal M\) and \(\theta\in\mathbb R\) be such that \(R_c(\theta)\) is strictly preperiodic and lands at the point~\(c\). Then \(R_M(\theta)\) also lands at~\(c\). (Note that the values of \(c\) for which the situation in the above theorem takes place, correspond to the so-called quadratic Misiurewicz polynomials.) The authors introduce a new method to prove Theorem~1. The main tool is the following result. \textbf{Theorem 2.} There exists a universal constant \(C_1>0\) such that \[ \sup_{z\in E}|S_\phi(z)|\rho_E(z)^{-2}\leq C_1 \sqrt{\mathcal D(\phi,E)} \] for any hyperbolic domain~\(E\) and any univalent function \(\phi:E\to\overline{\mathbb C}\). Here \(\rho_E\) stands for the Poincaré density on~\(E\) (density of the hyperbolic metric on \(E\)), \(S_\phi\) denotes the Schwarzian derivative of~\(\phi\), \[ S_\phi(z):=\frac{\phi'''(z)}{\phi'(z)}-\frac{3}{2}\left(\frac{\phi''(z)}{\phi'(z)}\right)^2, \] and \[ \mathcal D(\phi,E):=\sup\big|\mathrm {mod}(\phi(A))-\mathrm{mod}(A)\big|, \] where the supremum is taken over all doubly connected domains \(A\subset\overline{\mathbb C}\) of finite modulus such that \(\partial A\subset E\) and where (by abuse of notation) \(\phi(A)\) stands for the doubly connected domain bounded by \(\phi(\partial A)\).
scientific article; zbMATH DE number 5712103