Spherical hypersurfaces and Lattès rational maps (Q1267279)

A Lattès map is a rational map with an empty Fatou set. It is a chaotic rational map \(f\) on the Riemann sphere \(\widehat{{\mathbb C}}\) induced from an expanding complex multiplication \(s\) on some torus \(T\) by means of an elliptic function \(q: T\to\widehat{{\mathbb C}}\) that is a regular branched cover. These were discovered by S. Lattès in 1918. Let \(F\) be a nondegenerate homogeneous polynomial self-map of \({\mathbb C}^2\) that canonically induces \(f\) and let \(\Omega_F\) denote the basin of attraction for \(F\) at the origin. If \(f\) is a Lattès, map the authors show that the Brolin-Lyubich measure of \(f\) is smooth and strictly positive on some open set of \(\widehat{{\mathbb C}}\), that the intersection \(b\Omega_F\cap V\) (\(b\Omega_F\) is the boundary of the basin \(\Omega_F\)) is smooth and strictly pseudoconvex for some nonempty open set \(V\subset{\mathbb C}^2\), and \(b\Omega_F\) is spherical except on a finite union of circles that project onto a finite subset of \(\widehat{{\mathbb C}}\); in fact, these four conditions are all equivalent. The proof involves an investigation of the structure of the group \(G\) of automorphisms of \({\mathbb C}^2\) that preserve \(b\Omega_F\). A further study of this group \(G\) enables the authors to find examples of domains that are almost strictly pseudoconvex but do admit noninjective proper holomorphic maps.