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Spherical hypersurfaces and Lattès rational maps. (English) Zbl 0913.58031
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.
Reviewer: J.S.Joel (Kelly)

##### MSC:
 37B99 Topological dynamics 30D30 Meromorphic functions of one complex variable, general theory
##### Keywords:
rational maps; Fatou sets; basin of attraction; chaotic maps
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