A family of critically finite maps with symmetry
Abstract
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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- July 2003
- DOI:
- 10.48550/arXiv.math/0307057
- arXiv:
- arXiv:math/0307057
- Bibcode:
- 2003math......7057C
- Keywords:
-
- Dynamical Systems;
- 37F45
- E-Print:
- 24 pages, 9 figures