A family of critically finite maps with symmetry

Abstract

The symmetric group Sn acts as a reflection group on CP n−2 (for n ≥ 3) . Associated with each of the n 2 transpositions in Sn 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 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.

Dynamics of symmetric holomorphic maps on projective spaces

“…Crass [3] selected the symmetric group S k+2 as a finite group acting on P k and found an S k+2 -equivariant map which is holomorphic and critically finite for each k ≥ 1. We denote by C = C(f ) the critical set of f and say that f is critically finite if each irreducible component of C(f ) is periodic or preperiodic.…”

“…In particular g is critically finite. Although Crass [3] used this explicit formula to prove Theorem 1, we shall only use properties of the S k+2 -equivariant maps described below.…”

“…Let us look at properties of the S k+2 -equivariant map g on P k for a fixed k, which is proved in [3] and shall be used to prove our results. Let L k−1 denote one of the transposition hyperplanes, which is isomorphic to P k−1 .…”

“…Crass [2] extended Doyle and McMullen's algorithm to higher dimensions to solve sextic equations. Crass [3] found a good family of finite groups and equivariant maps for which one may say something about global dynamics. Crass [3] conjectured that the Fatou set of each map of this family consists of attractive basins of superattracting points.…”

“…Crass [3] found a good family of finite groups and equivariant maps for which one may say something about global dynamics. Crass [3] conjectured that the Fatou set of each map of this family consists of attractive basins of superattracting points. Although I do not know whether this family has relation to solving equations or not, our results will give affirmative answers for the conjectures in [3].…”