The symmetric group Sn acts as a reflection group on CPn-2 (for n>=3).Associated with each of the (n2) transpositions in Sn is an involution on CPn-2 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.
@article{urn:eudml:doc:41555,
title = {A family of critically finite maps with symmetry.},
journal = {Publicacions Matem\`atiques},
volume = {49},
year = {2005},
pages = {127-157},
zbl = {1116.37036},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41555}
}
Crass, Scott. A family of critically finite maps with symmetry.. Publicacions Matemàtiques, Tome 49 (2005) pp. 127-157. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41555/