A classification of bicritical rational maps with a pair of period two superattracting cycles
Let \(f\) and \(g\) be unicritical monic polynomials of degree \(d\) (that is, the form \(z \longmapsto z^d+c,\) for some \(c\)). Denote by \(\widetilde{\mathbb{C}}=\mathbb{C} \cup \Big\{\infty\cdot e^{2 \pi i t},~~t \in \mathbb{R}/\mathbb{Z}\Big\}\) the complex plane compactified by the circle at infinity. We continuously extend the two polynomials to the circle at infinity by \[ f(\infty\cdot e^{2 \pi i t})=g(\infty\cdot e^{2 \pi i t})=\infty\cdot e^{2 d \pi i t}. \] The extended dynamical plane of \(f\) (respectively \(g\)) is denoted by \(\widetilde{\mathbb{C}}_f\) (respectively \(\widetilde{\mathbb{C}}_g\)). We define a topological sphere \(\Sigma_{f,g}\) by gluing the two extended planes together along the circle at infinity, that is, \[ \Sigma_{f,g}= \widetilde{\mathbb{C}}_f \cup \widetilde{\mathbb{C}}_g/\sim, \] where \(\sim\) is the relation which identifies the point \(\infty\cdot e^{2 \pi i t} \in \widetilde{\mathbb{C}}_f \) with the point \(\infty.e^{-2 \pi i t} \in \widetilde{\mathbb{C}}_g\). The formal mating is defined to be the branched covering \(f \uplus g:\Sigma_{f,g} \longrightarrow \Sigma_{f,g}\) such that \(f \uplus g|_{\widetilde{\mathbb{C}}_f}=f\) and \(f \uplus g|_{\widetilde{\mathbb{C}}_g}=g\). Assume that the filled Julia set \(K_h\) of a monic polynomial \(h :\widetilde{\mathbb{C}} \rightarrow \widetilde{\mathbb{C}}\) of degree \(d \geq 2\) is connected. Then, by Böttcher's theorem, there is a conformal isomorphism \[ \phi=\phi_h:\widetilde{\mathbb{C}}\setminus \overline{\mathbb{D}} \rightarrow \widetilde{\mathbb{C}}\setminus K_h \] which can be chosen so that it conjugates \(z \longmapsto z^d\) on \(\widetilde{\mathbb{C}}\setminus \overline{\mathbb{D}}\) with the map \(h\) on \(\widetilde{\mathbb{C}}\setminus K_h\). The external ray of angle \(t\) is given by \(R_h(t)=\phi_h(r_t),\) where \(r_t=\{r\exp(2\pi i t), r>1\}\subset \mathbb{C}\setminus \mathbb{D}\). We denote by \(\sim_h\) the smallest equivalence relation on \(\widetilde{\mathbb{C}_h}\) such that \(x \sim_h y\) if and only if \(x,y \in \overline{R_h(t)}\) for some \(t\). We further define the ray-equivalence relation \(\approx\) on \(\Sigma_{f,g}\) as the smallest equivalence relation on \(\Sigma_{f,g}\) generated by \(\sim_f\) and \(\sim_g\). We say that a rational map \(F\) is the geometric mating of \(f\) and \(g\) if \(F\) is topological conjugate via an orientation-preserving homeomorphism \(\phi:\Sigma_{f,g} \rightarrow \Sigma_{f,g}\) which is holomorphic on the interior of \(K_f \uplus K_g\). In the paper under review, the authors prove the following theorem. Theorem. A degree-\(d\) bicritical rational map with labeled critical points and with two periodic superattracting cycles may be realized as a mating in precisely \(d-1\) ways. Furthermore, the rational map is completely defined (up to Möbius conjugacy) by one piece of combinatorial data.
scientific article; zbMATH DE number 6167096