Resumen de A family of critically finite maps with symmetry
Scott Crass
The symmetric group $\mathcal{S}_{n}$ acts as a reflection group on $\mathbf{CP}^{n-2}$ (for $n\geq 3$). Associated with each of the $\binom{n}{2}$ transpositions in $\mathcal{S}_{n}$ is an involution on $\mathbf{CP}^{n-2}$ that pointwise fixes a hyperplane -the mirrors of the action. For each such action, there is a unique $\mathcal{S}_{n}$-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.
