A family of critically finite maps with symmetry

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.


Overview
Complex dynamics in several dimensions has been the object of considerable recent study. Some specialized previous work in this field treats a variety of maps that share a common property: they respect the action of a finite group on a complex projective space. (See [C1], [C2], [C3].) The nature of these investigations leads to a consideration of issues pertaining to global dynamics. While the most significant dynamical claims possess experimental support, they remain theoretical conjectures. The current project stems from a desire to find symmetrical maps with interesting geometry and tractable dynamics. Its first fruit is an infinite family of special maps each of whose members respect the action of the symmetric group S n . In fact, for each n ≥ 3, there is a unique holomorphic map g on CP n−2 whose critical set consists of an S n orbit of n 2 hyperplanes that g preserves. This leads to a strong form of critical finiteness that yields several global dynamical results of the type that eluded earlier undertakings.
The treatment develops in three stages: (1) some background on special actions of S n and their associated symmetrical maps (2) proofs that the special family of critically-finite maps with S n symmetry exists and that each member is unique and holomorphic (3) proofs of claims concerning the dynamics of the maps (in the cases n = 3, 4) Specifically, each member has a certain attractor with dense basins. When n > 4, the claim concerning the attractor is conjectured.
Finally, some graphical results for low-dimensional cases appear.
2. S n acts on CP n−2 The permutation action of the symmetric group S n on C n preserves the hyperplane H = n k=1 x k = 0 ≃ C n−1 and, thereby, restricts to a faithful (n − 1)-dimensional irreducible representation. This action on C n−1 projects one-to-one to a group G n on H := PH ≃ CP n−2 .
2.1. Special orbits and reflection hyperplanes. The smallest G n orbit consists of the n points Note that the null space of A is the euclidean orthogonal complement to H. This change of coordinates has an "inverse" which gives AB = −n I n−1 BA = 1 n − n I n where I m is the m×m identity and 1 n is the n×n matrix each entry of which is 1. Accordingly, A and B induce isomorphisms between H and CP n−2 .
In u-coordinates, the n 2 -hyperplanes are the n−1 coordinate hyperplanes {u k = 0} and the n−1 2 spaces {u k = u ℓ }. The points determined by the intersections of the n 2 -hyperplanes play a central role in subsequent developments. Their description is especially simple in u. (See Table 1.) With one exception, each orbit consists of points p k and q k with complementary coordinates.
representative points n on n − 2 hyperplanes orbit size Relative to the u space, G n is generated over the permutation action G n−1 of S n−1 on the u k by means of the involution that transposes the pair {p 1 , q 1 } and fixes the remaining members of the n-point orbit. Note that T is the u version of the transformation

G n equivariants
Consider a map f = [f 1 , . . . , f n−1 ] from H to itself given by homogeneous polynomials in u = (u 1 , . . . , u n−1 ) of degree r. In general, f can be meromorphic; that is, for some p ∈ C n−1 , f (p) = 0 for every lift of f to C n−1 . We say that f is G n -equivariant when it sends a group orbit to a group orbit. Algebraically, this means that f commutes with every element of G n . Obviously, f is G n−1 -equivariant as well. It readily follows that each component f k is invariant under the stabilizer Z k of u k . Thus, we can express a component by where A k,ℓ is a degree-ℓ Z k invariant. Accordingly, each A k,ℓ is taken to be a polynomial in the elementary symmetric functions in the complementary variables u k = (u 1 , . . . , u k−1 , u k+1 , . . . u n−1 ). Alternatively, we can employ the elementary symmetric functions in u when expressing A k,ℓ . This is a matter of expressing a polynomial in u k in terms of a polynomial in u and a polynomial in u k with lower degree. Specifically, let S m and S m be the degree-m elementary symmetric functions in u k and u respectively. Taking S 0 = 1, the relations give a reductive scheme for the replacement process.
An immediate consequence of G n−1 equivariance is that We can say a bit more concerning the form that G n equivariants take. First, consider a point a that some element M ∈ G n fixes and observe that M f (a) = f (M a) = f (a). Thus, f either sends a to another fixed point of M or blows up at a-that is, for any lift f and a of f and a to C n−1 , f ( a) = 0. Applying this condition to the n 2 -hyperplanes, provided that a is not a point of indeterminacy, each point on such a hyperplane must map to a point that is fixed by the involution that fixes the hyperplane pointwise. The only place for the image of such a point is on the hyperplane itself or its companion point. Under a holomorphic map, the image cannot be the companion point-this would force the entire hyperplane to collapse to the point. So, a holomorphic G n equivariant f sends an n 2 -hyperplane to itself. This circumstance forces f k to be divisible by u k and, thereby, requires the terms A r to be a power of S n−1 or to vanish. In particular, when r ≤ n − 1, A r = 0 so that By design, the map f has G n−1 symmetry. To be fully G n -equivariant, the map must commute with T as well. This condition places strong restrictions on the A ℓ . The general form they take might be an interesting result, but not one taken up by the current investigation. Here, the quest is for a family of G n equivariants with very special properties.

Reflection hyperplanes as critical sets: existence, uniqueness, and holomorphy
Explicit computation in low-degree cases reveals the existence of a unique holomorphic G n equivariant whose critical set is precisely the n 2 -hyperplanes counted with multiplicity two. These maps conform to a general formula. Let g = [g 1 , . . . , g n−1 ] where and S n,ℓ is the degree-ℓ elementary symmetric function in u 1 , . . . , u n−1 . In the degree-0 case, take S n,0 = 1. By construction, each g is equivariant under the group G n−1 that permutes the u ℓ . In addition, the u 3 ℓ factor in each coordinate implies that the maps are doubly critical on n − 1 of the n 2 -hyperplanes-namely, where u ℓ = 0. Were g to commute with the transformation T that generates G n over G n−1 , symmetry would provide for double criticality on the remaining n−1 2 of the n 2 -hyperplanes-where u j = u k . Moreover, since a degree-(n+1) map in n − 1 variables has a critical set whose degree is (n − 1) n = 2 n 2 , g's critical set would consist exclusively of the n 2 -hyperplanes. This section develops rather technical arguments for three main results. According to Theorem 4.1, the n 2 -hyperplanes form g's critical set with multiplicity two. Moreover, Theorem 4.2 informs us that there is only one such map for each G n action. Theorem 4.3 states that each g is holomorphic on H which implies that g preserves each n 2 -hyperplane L rather than collapse L to a lower-dimensional variety; a contraction would force the map to blow up. Thus, g is a family of maps each member of which is holomorphic, doublycritical on the n 2 -hyperplanes, and critically-finite. As a standing assumption, let n ≥ 3.
Theorem 4.1. The respective g is T -equivariant, hence, G n -equivariant.
Theorem 4.2. Under the action of G n , g is the unique rational map of degree n + 1 for which each n 2 -hyperplane is doubly critical. Theorem 4.3. Each member of the family g is holomorphic on H.
Proof of Theorem 4.1. Propositions 4.5 and 4.8 below establish that g is symmetric under T as well as under G n−1 . Since T generates G n over G n−1 , g is G n -equivariant.
The proofs of the propositions rely on a formula that describes how the elementary symmetric functions transform under T . This result was found by pattern detection in low-degree cases. For simplicity of appearance, express the functions S n,k (u) in the suppressed form S n,k .
Lemma 4.4. For k ≤ n, the G n−1 invariants S n,k transform under T according to Proof. Proofs of several technical lemmas appear in the appendix.
The argument for the T -equivariance of g examines the coordinates individually.
Proposition 4.5. The factor G 1 of g 1 is T -invariant (in a linear as well as projective sense).
Proof. The proof amounts to manipulation of sums. Since n is fixed here, let S k = S n,k . Consider By Lemma 4.4, Setting m = k + ℓ, Reversing the order of summation, Lemma 4.6 below gives the sum over k: Proof. See the appendix.
Corollary 4.7. Each g is T -equivariant in the first coordinate.
Proof. Let [·] 1 specify a map's first coordinate. Then To establish overall T -equivariance, it suffices to consider the behavior of g under T in just the second coordinate. This follows directly from the commutativity of T and the members τ 2,m ∈ G n that simply transpose the second and mth basis elements: provided that m = 1, n. Expressed in terms of S n , this amounts to the commutativity of the disjoint transpositions (1n) and (2m). So, noting that g is G n−1 -equivariant, hence, τ 2,m -equivariant, and given that g is Tequivariant in its second coordinate, Proposition 4.8. The second coordinate of g satisfies the equivariance condition Proof. First, express g 2 • T in a way that's useful for comparison to [T • g] 2 .
Again, set S k = S n,k . Applying Lemma 4.4, Reversing the order of summation, Lemma 4.9 below establishes a useful identity for the sum over k so that The first two terms are g 2 (u) and g 1 (u) respectively. Since their difference amounts to [T g(u)] 2 , Adding and subtracting −u 1 u 2 S n−1 on the right, Let m = p − 1, while, for the apparent variables u 1 and u 2 , set x = u 2 and y = u 1 . The result is Thus, when x = u 2 and y = u 1 , Proof. See the appendix.
Now we turn to the matter of uniqueness.

Proof of Theorem 4.2. Suppose that
is a map of this type. The strategy is to compare g to h in terms of u coordinates. Since h is G n−1 -equivariant and doubly critical on each {u k = 0}, the components of h have the form h k = u 3 k H k . Furthermore, each H k is a degree-(n−2) invariant under an S n−2 -isomorphic subgroup of G n−1 , namely, the stabilizer of u k . It follows that we can express these polynomials by By G n−1 symmetry, we can examine a single component: h 1 , say. Now, consider V n−2 . In the event that u 1 divides V n−2 , the associated component takes the form h 1 = u 4 1 H 1 . But this implies that {u 1 = 0} is triply critical which is at odds with the assumption that h is doubly critical on the n 2 -hyperplanes. By degree counting, the latter state of affairs completely accounts for the critical set.
Accordingly, assume that V n−2 ≡ 0 when u 1 = 0. We can now say that The G n−1 invariance of V n−2 requires that every element in the G n−1 orbit of Y appears in V n−2 and only these terms appear. Thus, Recalling the form of g, lift g and h to maps g and h on C n−1 so that Also, we can lift G n−1 trivially to a linear group G. Consequently, the G equivariant g − h is either the zero map or is both doubly critical along the n 2 -hyperplanes and, as in the case considered above, has the contrary property that its first component is divisible by u 4 1 . Hence, the former case is the only possibility so that h = g.
Evidently, g's uniqueness is due to its full G n symmetry-that is, to its T -equivariance in addition to its symmetry under G n−1 . The proof of the following lemma makes this explicit.
Proof. Let k = 1. Equivariance under T requires the components of h to satisfy the following identities: (To lessen clutter, suppress explicit mention of the variable u, where possible.) By (1), Turning to (2), Expanding the first binomial on the left, using (3), and rearranging gives Dividing through by the common factor u k − u 1 , Note that this expression makes sense since n ≥ 3. Thus, The upcoming proof of Theorem 4.3 exploits a dimension-reducing process of restricting g to intersections of n 2 -hyperplanes. This cascade of intersections leads to the special point-orbits determined by the hyperplanes. At these points, the map's behavior is explicitly computable.
Proof of Theorem 4.3. When n = 3, g is one-dimensional and hence, holomorphic. As for the non-trivial cases n > 3, choose the "literal" lift of g to C n−1 : g = (g 1 , . . . , g n−1 ) where Let X denote the union of the n 2 -hyperplanes lifted to hyperspaces through 0 in C n−1 . Suppose there is a point a ∈ C n−1 where g(a) = 0. By homogeneity, is the Jacobian matrix of g. Thus, g is critical at a. That is, a is a zero eigenvector for D g(a). In this case, the map collapses in the "radial" direction defined by a. Since g is critical only on X, a lies on one of X's constituent hyperspaces; call this hyperspace L n−2 (≃ C n−2 ) and consider the restriction g n−2 of g to L n−2 . (Note that the action of G n restricted to L n−2 is isomorphic to S n−2 so that g n−2 is not the member of the family g for dimension n − 2 where the action is that of S n−1 .) Since g n−2 (a) = 0, a is a zero eigenvector for D g n−2 (a); the critical set of g n−2 contains a. But, a zero eigenvector v for D g n−2 (a) corresponds to a radial collapse in the v direction so that v is also a zero eigenvector for D g(a). But, as Lemma 4.11 below describes, det D g n−2 (a) does not vanish identically on L n−2 so that the critical set of g n−2 is a proper algebraic subset of X and L n−2 . Hence, the only possible location for a is where some hyperspace in X different from L n−2 intersects L n−2 . Denote this intersection by L n−3 . Further reducing the dimension, let g n−3 = g| L n−3 so that g n−3 (a) = 0 and a is critical for g n−3 . As above, a belongs to the intersection of L n−3 with a hyperspace in X that does not contain L n−3 . This reduction continues with the outcome at each stage that a belongs to the intersection of n 2 -hyperplanes. When the procedure arrives at dimension three, a lies on two planes through 0 in C n−1 -that is, a point in CP n−2 -that are intersections of n 2 -hyperspaces. But, Lemma 4.12 below implies that g = 0 at these points.
Lemma 4.11. For the restriction g of g to any space L m of dimension m = 0 determined by the intersection of hyperspaces in X, det D g ≡ 0.
Proof. By the permutation action of G n−1 on the u k , we can take Any L m space that is partially determined by the intersection of p sets of the form {u k = 0} belongs to the G n−1 -orbit of the set specified above. Relabel the coordinates on L m so that the restriction is expressed Let D g i be the Jacobian of g i so that In order for det D g ≡ 0, the set {D g i , i = 1, . . . , m} must be linearly dependent in functional terms. To establish linear independence, consider the relation  It suffices to compute G 1 (p m ). A straightforward calculation gives With this, From Lemma 4.13 below, Proof. See the appendix.

Reflection hyperplanes as critical sets: Global dynamics
Let L n−3 generically denote an n 2 -hyperplane and let X refer to the union of the L n−3 . Where m of the L n−3 intersect to form a CP n−2−m , call the resulting space L n−2−m . (Note that more than m of the L n−3 can pass through an L n−2−m .) Not only is g critically-finite on H ≃ CP n−2 with critical set consisting of the L n−3 hyperplanes, the restriction g| L n−2−m is also critically-finite, having a collection of the L n−3−m for its critical set. In [FS1], such behavior is called strict critical finiteness (Section 7). In fact, all of the L n−3−m on an L n−2−m are critical for g| L n−2−m though not with the same multiplicity.
5.1. The Fatou set of g. Following standard practice, the Fatou set F g is where the family of iterates {g k } is normal and the Julia set J g is the complement of F g .
The behavior of g on an L n−3 plays a central dynamical role. Again, lift g to C n−1 : g = (g 1 , . . . , g n−1 ) with For a space L m ⊂ CP k lifted to C k+1 , call the lifted space L m+1 .
Proposition 5.1. For any a ∈ L n−3 , g is critical in the direction off of the hyperplane.
Proof. By symmetry, consider the L n−2 given by {u 1 = 0}. For any a ∈ {u 1 = 0}, the first row of D g(a) vanishes. Thus, the local behavior of g collapses points onto L n−2 . Explicit calculation reveals that the collapse occurs in the direction of (2, 1, . . . , 1).
Recall that the p m represent the point sets of G n orbits determined by intersecting the L n−3 . Refer to these orbits as "p m -points." First of all, each such point is superattracting in all directions.
Theorem 5.2. Under g, the fixed p m -points are superattracting in every direction. Conversely, the only points that are superattracting in every direction are the p m -points.
Proof. To establish that, at p m , g is critical in every direction in CP n−1 show that the Jacobian D g at p m has rank 1. Here, p m is lifted in the literal way. It then follows that, since g(p m ) = 0, there are n − 2 non-radial directions through p m that have zero eigenvalue.
The Jacobian has the form With S k = S n,k a straightforward calculation establishes that, for ℓ ≤ m, By manipulation of sums, The second sum is ∂G j ∂u ℓ (p m ) for j, ℓ ≤ m and j = ℓ. To show that the first sum amounts to −3 G i (p m ), notice that, from the proof of Theorem 4.3, Finally, the calculation reduces to showing that the first sum vanishes. This follows readily by splitting the sum into two terms each of which is a binomial expansion of 1 − 1. Specifically, Thus, the nonzero rows of D g(p m ) are identical and the matrix has rank 1. For the converse claim, consider a point q that is critical in every direction. When g is restricted to any intersection L k of hyperplanes each of which is an L n−2 , q is again critical for the restriction g| L k . Hence, q lies on some L n−2 that does not contain L k and so, is determined by the intersection of L n−2 spaces. Now for the issue of the Fatou set F g . Is there a Fatou component of g that is not in the basin of a p m point? Theorem 5.3. For n = 3, 4, F g consists of the basins of attraction of the p m -points.
Proof. When n = 3, the one-dimensional map g has three fixed critical p mpoints. A basic result in one-dimensional dynamics states that the Fatou set of a rational map with periodic critical points consists only of superattracting basins; indeed, the basins have full measure in CP 1 .
In the two-dimensional case n = 4, the claim follows from Theorem 5.2 and [FS1], Theorem 7.7. The latter implies that if a holomorphic map f on CP 2 has a critical set C such that 1) C is periodic and 2) CP 2 − C is Kobayashi hyperbolic, then f has only superattracting basins in its Fatou set. See below for an explanation of the fact that condition 2) applies to g.

The general case remains open.
Conjecture 5.4. For n ≥ 5, F g consists of the basins of attraction of the p m -points.
One approach to this claim adopts a technique from the proof of Theorem 4.3: reduction of dimension to the one-dimensional case where some things are understood. The argument for Theorem 5.6 employs the same idea. Assume an arbitrary choice of n ≥ 5.
The question of whether the basins of the p m -points exhaust F g calls for some preparation. Following [U], let C f be the critical set of a holomorphic map f on CP m , be the postcritical set and the ω-limit set of C f respectively. Also, the Fatou limit set Λ f is where the forward orbits of Fatou components accumulate. In the case of g, D g = E g = X.
Let p ∈ F g and U be the Fatou component to which p belongs. For a critically-finite map f , Λ f ⊂ E f ( [U], Theorem 5.1). Accordingly, the forward orbit {g k (p)} of p accumulates on some L n−3 and, by Proposition 5.1, is attracted to that L n−3 -call it L n−3 as well. Accordingly, The claim also follows from [M], Theorem 2.36-a result established by consideration of expansion in the Kobayashi metric on the complement of the postcritical set.
The task now is to show that g r (U ) ∩ L n−3 = ∅ for some r. An argument might develop in two steps: 1) the orbit of a point that is Fatou for g accumulates at points that are Fatou for g := g| L n−3 ; 2) a point that is Fatou for g is also Fatou for g and, thereby, belongs to a Fatou component in CP n−2 .
To treat the first claim, let q ∈ L n−3 be a limit point of {g k (p)} with g n k | K −→ h where h : K → L n−3 , K ⊂ U is a neighborhood of p, and h(p) = q.
Suppose that q belongs to the Julia set J g . By Proposition 5.1 g is superattracting at g k (q) in some direction away from L n−3 for all k. This equips q with a stable set transverse to L n−3 . If g were hyperbolic-as in the case n = 4, one might expect that the Kobayashi expansion at q would produce saddle-like behavior and force U to contain Julia points for g.
To see claim 2) above, let q ∈ F g with a neighborhood N on which { g k } is normal. Take N to be the connected neighborhood of q that is absorbed by N and includes N ; that is, N is the connected component of the stable where N ⊂ N and S x is the stable set of x. Every point in N belongs to some S x . Thus, if g n k converges to h on N , then g n k converges on N to The claims 1) and 2) imply that some g r (U ) intersects L n−3 ; indeed, g r (U ) ∩ L n−3 is a Fatou component for g. By the critical finiteness of g, the forward orbit of g r (U ) ∩ L n−3 meets some L n−4 in Fatou points for g| L n−4 . This cascade continues until some g s (U ) makes contact with a line L 1 , in particular, with the Fatou set of g| L 1 . Since g| L 1 has fixed critical points, it has only superattracting basins. The only critical points on L 1 are p mpoints. Hence, g s (U ) ∩ L 1 lies in the basin of attraction of some such point.
How "large" are the basins of the p m -points? First of all, let be the precritical set of f . The following basic result yields that the closure of B g contains the Julia set J g . ( [FS1], Proposition 6.5.) Theorem 5.5. If f : CP k → CP k is holomorphic and CP k − B f is hyperbolically embedded, To apply this result to g, we must see that it satisfies the hypotheses. By Theorem 4.3, g is holomorphic on CP n−2 . Two theorems of M. Green imply that L n−1−m − B g| L n−1−m is hyperbolically embedded in L n−1−m (taking L n−2 = H). (For details on Green's results, consult [FS1], Section 5.) To see this, suppose that, for n ≥ 4 and m ≥ 2, φ : C −→ L n−1−m − B g| L n−1−m is holomorphic. Then φ(C) omits at least n−m+1 hypersurfaces in L n−1−m , namely, some L n−m−2 spaces and their preimages. By one of Green's theorems (Theorem 5.6 in [FS1]), φ(C) is contained in a compact complex hypersurface. Since such a hypersurface intersects the omitted hypersurfaces, φ(C) omits at least three points and so, is constant. The statement concerning hyperbolic embedding follows from Green's other theorem (Theorem 5.5 in [FS1]).
One other preliminary: since C g ⊂ g −1 (C g ), J g ⊂ A g = B g . We can now establish a bit of F g 's global structure.
Theorem 5.6. Under the assumption that Conjecture 5.4 holds, the Fatou set F g is dense in H.
Proof. Consider j 0 ∈ J g and let U 0 be a neighborhood of j 0 . By Theorem 5.5, some precritical points meet U 0 so that, for some m,

If
U 1 := g m (U 0 ) ∩ L n−3 fails to contain Julia points, the case is made. Otherwise, take a Julia point j 1 ∈ U 1 , a neighborhood of j 1 .
The map g| L n−3 is critically finite with critical set C n−3 in the intersection of L n−3 and the hyperplanes in X different from L n−3 . Hence, C n−3 is a collection of L n−4 spaces. Implementing the argument given for j 0 and U 0 under g using j 1 and U 1 under g| L n−3 produces a neighborhood of a Julia point j 2 on some L n−4 . The descent continues until it reaches a Julia point j n−3 and neighborhood U n−3 on an L 1 . Thus, U n−3 meets the Fatou set of g| L 1 . Since g| L 1 has fixed critical points that are p m -points, its Fatou set consists of the superattracting basins of those p m -points. Accordingly, U n−3 -hence, U 0 -contains points in F g .

5.2.
A query on the structure of g's Julia set. For the restricted map g = g| L n−3 , the Julia set is given by The inclusion J g ⊂ J g ∩ L n−3 is clear. If x / ∈ J g , then x belongs to a basin of a p m -point so that x / ∈ J g . At each point p ∈ J g , the map is superattracting in the direction away from L n−3 . Thus, there is a "stable set" S p of points in J g whose orbits are attracted to the orbit of p. Accordingly, there is a stable bundle over J g S J g := p∈J g S p ⊂ J g .
Are the S p one-dimensional manifolds? Are the preimages of the S J g dense in J g ? In the case n = 4, g restricts to a critically-finite map g on an L 1 that is one of the six lines of reflection for G 4 . Figure 4 displays the three basins of attraction for g. The Julia set J g consists of the boundaries of these basins. For each Julia point p ∈ L 1 , there is an S p away from the line. What can be said about the structure of S J g ?
What about the points that are not absorbed by X?
On an L 1 , each Julia point is non-wandering and has a contracting direction onto L 1 and an expanding direction in L 1 . For a hyperbolic map on CP 2 , the literature describes a grading of the non-wandering set Ω by the expanding dimension ( [FS2]): The p m -points comprise Ω 0 and ∪ X J g ⊂ Ω 1 . The non-wandering points not on X belong to K. Since any neighborhood of such a point p contains an open set that is attracted to X, there is expansion at p. Does it happen that Ω ∩ K ⊂ Ω 2 so that g is hyperbolic?
6. Geometry and dynamics in low-dimension To avoid confusion, let g n+1 represent the particular map g on the respective G n -symmetric H.
6.1. The one-dimensional case: g 4 and Halley's method. When n = 3, the reflecting "hyperplanes" consist of a three-point orbit. With these points located at {1, ρ, ρ 2 | ρ = e 2 π i/3 }, the map's inhomogeneous expression on {u 2 = 0} is We can realize the G 3 action on CP 1 by the polyhedral configuration of a double triangular pyramid-two regular tetrahedra joined at a face. The two-point orbit resides at 0 and ∞ and defines two hemispheres in the usual way. Accordingly, the unit circle corresponds to the equatorial boundary between hemispheres and the 3-points {1, ρ, ρ 2 } are vertices where four faces congregate.
Consider the degree-4 map that fixes the vertices of each face and sends one face F to four others: F itself and the three faces in the hemisphere not containing F. This symmetrical construction results in G 3 -equivariant behavior. At the three equatorial vertices, the map opens up a face's internal angle of π/2 to an angle of 3 π/2 so that the local behavior is cubing. This makes the 3-point orbit doubly-critical and, by degree counting, the entire critical set. Accordingly, this map must be g 4 . Since g 4 has periodic critical points, the superattracting basins constitute its Fatou set and, moreover, have full measure in CP 1 . A portrait of the basins appears in Figure 1.
It turns out that g 4 is Halley's Method-a variation on Newton's Methodfor a cubic polynomial. (See [ST] for a description of Halley's Method in real variables.) In the coordinates selected above, the polynomial to which we apply Halley's method is 6.2. The map in two dimensions. Since g n+1 has real coefficients, it preserves the RP n−2 of points whose coordinates can be expressed by real numbers. Call this space R. Under G 4 , R has the structure of a projective cube. We can view this as a hemisphere where one vertex is at the pole and the other three vertices lie along a circle whose center is the distinguished vertex. The 3-point orbit (i.e., the face-centers) lies on another circle centered at the north pole. 1 − 10 x 2 + 20 x 3 + 30 x 4 + 40 x 5 − 10 y 2 − 60 x y 2 + 60 x 2 y 2 − 80 x 3 y 2 + 30 y 4 − 120 x y 4 , 24 −5 x y 3 + 5 x 2 y 3 + y 5 1 − 10 x 2 + 20 x 3 + 30 x 4 + 40 x 5 − 10 y 2 − 60 x y 2 + 60 x 2 y 2 − 80 x 3 y 2 + 30 y 4 − 120 x y 4 .
The six lines of reflection run along the edges and a diagonal of a face. These lines carve the hemisphere into twelve triangles each of which is a fundamental domain for the reflection group action G 4 . Viewing the "hemicube" from above an edge, Figure 3 reveals the map's action on a fundamental triangle: one triangle stretches and twists onto five other associated triangles. (where {u 1 = u 2 } intersects {u 3 = 0}). In coordinates where the two-point orbit is ±1 and the one-point orbit is 0, the map takes the form z −→ 4 z 3 (z 2 + 5) 15 z 4 + 10 z 2 − 1 . Figure 4 shows their basins of attraction on the line. Notice that this CP 1 intersects R in an RP 1 that corresponds to a line of reflective symmetry in Figure 2 and the horizontal mirror in Figure 4-for instance, the line that passes through the red, gray, and yellow basins.  (Figure 5), the geometry is that of a projective double triangular pyramid and these points respectively occupy The map is given by z −→ 8 z 3 (3 z 2 + 5) 5 z 6 + 45 z 4 + 15 z 2 − 1 and z −→ 8 z 4 (z 2 − 2 z + 5) 5 z 6 + 30 z 5 + 15 z 4 + 20 z 3 − 5 z 2 − 2 z + 1 . Figure 5. Dynamics of g 6 on the S 3 -symmetric RP 2 As before, each CP 1 intersects the RP 2 of Figure 5 in an RP 1 : the three lines {u k = 0 | k = 1, 2, 3} give the edges of the "triangle" whose vertices are (1, 0), − 1 2 , ± √ 3 2 and the three lines {u k = u ℓ | k, ℓ = 1, 2, 3} correspond to the lines of reflective symmetry through (0, 0). Figure 6. Dynamics of g 6 on the Z 2 -symmetric CP 1 Figure 7. Dynamics of g 6 on the Z 1 -symmetric CP 1
Lemma (4.6). Proof. Consider the expansion of the generating function Now, differentiate and evaluate at x = 1: Rearranging this equation yields the desired statement.