A family of critically finite maps with symmetry

Details A family of critically finite maps with symmetry A family of critically finite maps with symmetry

2003-07-03

arxiv.org/abs/math/0307057v2

Mathematics

Dynamical Systems

24 pages, 9 figures

Scientific paper

The symmetric group S_n acts as a reflection group on CP^{n-2} (for $n\geq 3$) . Associated with each of the $\binom{n}{2}$ transpositions in S_n is an involution on CP^{n-2} that pointwise fixes a hyperplane--the mirrors of the action. For each such action, there is a unique S_n-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.

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