A Family of Critically Finite Maps with Symmetry

The symmetric group Sn acts as a reflection group on CP n−2 (for n ≥ 3) . Associated with each of the ( n 2 ) transpositions in Sn is an involution on CP that pointwise fixes a hyperplane—the mirrors of the action. For each such action, there is a unique Sn-symmetric holomorphic map of degree n+1 whose critical set is precisely the collection of hyperplanes. Since the map preserves each reflecting hyperplane, the members of this family are critically-finite in a very strong sense. Considerations of symmetry and critical-finiteness produce global dynamical results: each map’s Fatou set consists of a special finite set of superattracting points whose basins are dense. 1. Overview Complex dynamics in several dimensions has been the object of considerable recent study. Some specialized previous work in this field treats a variety of maps that share a common property: they respect the action of a finite group on a complex projective space. (See [C1], [C2], [C3].) The nature of these investigations leads to a consideration of issues pertaining to global dynamics. While the most significant dynamical claims possess experimental support, they remain theoretical conjectures. The current project stems from a desire to find symmetrical maps with interesting geometry and tractable dynamics. Its first fruit is an infinite family of special maps each of whose members respect the action of the symmetric group Sn. In fact, for each n ≥ 3, there is a unique holomorphic map g on CP whose critical set consists of an Sn orbit of ( n 2 ) hyperplanes that g preserves. This leads to a strong form of critical finiteness that yields several global dynamical results of the type that eluded earlier undertakings. The treatment develops in three stages: (1) some background on special actions of Sn and their associated symmetrical maps (2) proofs that the special family of critically-finite maps with Sn symmetry exists and that each member is unique and holomorphic Date: February 8, 2008. 2000 Mathematics Subject Classification. Primary 37F45 Secondary 20C30.