A family of critically finite maps with symmetry
- Autores: Scott Crass
- Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 49, Nº 1, 2005, págs. 127-157
- Idioma: inglés
- DOI: 10.5565/publmat_49105_06
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Resumen
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.
