On biaccessible points in the Julia sets of some rational functions
Let \(\Omega\subset\widehat{\mathbb C}:=\mathbb C\cup\{\infty\}\) be a simply connected domain such that \(\partial \Omega\) contains more than one point. An external ray in \(\Omega\) is the image of a ray \(\ell_t:=\{re^{it}:r>1\}\), \(t\in\mathbb R\), under the conformal mapping \(\Phi:\infty\mapsto\infty\) of~\(\{z\in\hat{\mathbb C}:|z|>1\}\) onto~\(\Omega\). An external ray \(R\) is said to land at a point \(z_0\in\partial\Omega\) if \({\text{cl}}(R)\cap\partial\Omega=\{z_0\}\). A point \(z_0\in\partial \Omega\) is said to be biaccessible from \(\Omega\) if there exist two different external rays that land at~\(z_0\). It is known that \(z_0\in\partial\Omega\) is biaccessible from \(\Omega\) if and only if \(z_0\) is a cut-point of \(\partial\Omega\), i.e., if \(\partial\Omega\setminus\{z_0\}\) is not connected. The paper is devoted to biaccessible points of immediate basins of attraction~\(\mathcal A_\infty\) for rational functions having a superattracting point at~\(\infty\) and an irrationally indifferent fixed point in~\(\mathbb C\). The interest in the biaccessible points is motivated by the fact that for polynomials, \(\partial \mathcal A_\infty\) coincides with the Julia set. As an extension of a result from [\textit{S. Zakeri}, ``Biaccessibility in quadratic Julia sets'', Ergodic Theory Dyn. Syst. 20, No.6, 1859--1883 (2000; Zbl 0970.37037)] the author proves the following theorem: Let \(f\) be of the form~\(f(z)=z^d+c\), \(c\in\mathbb C\), \(d\in\mathbb N\), \(d\geq2\). Suppose that \(f\) has an irrationally indifferent fixed point~\(\alpha\) and that \(\mathcal A_\infty\) has a biaccessible point~\(z_0\). If \(\alpha\) is a Siegel point, then \(0\in \mathcal O^+(z_0):=\{f^{\circ n}(z_0):n=0,1,\ldots\}\). If \(\alpha\) is a Cremer point, then either \(0\in\mathcal O^+(z_0)\) or \(\alpha\in\mathcal O^+(z_0)\). A similar result is obtained for the family \(g_\theta(z):=e^{2\pi i\theta}z+z^d\), \(\theta\in\mathbb R\): if the origin is a Siegel point of \(g_\theta\), or, alternatively, the origin is a Cremer point of~\(g_\theta\) but does not belong to~\(\mathcal O^+(z_0)\), then at least one of the critical points of~\(g_\theta\) is contained in~\(\mathcal O^+(z_0)\). Finally, for the family \(h_{\theta,a}(z):=e^{2\pi i\theta}z^2(z-a)/(1-\bar a z)\), \(\theta\in\mathbb R\), \(|a|>3\), of finite Blaschke products, the author proves that if the rotation number of \(h_{\theta,a}|_{|z|=1}\) is irrational and \(\mathcal A_\infty\) has a biaccessible point~\(z_0\), then \(\mathcal O^+(z_0)\) contains the critical point of~\(h_{\theta,a}\) that lies in~\(\{z:|z|>1\}\).
scientific article; zbMATH DE number 5708832