A family of critically finite maps with symmetry

Publ. Mat. 49 (2005), 127–157 A FAMILY OF CRITICALLY FINITE MAPS WITH SYMMETRY Scott Crass Abstract The symmetric group Snacts as a reflection group on CPn−2 (for n≥3) . Associated with each of the `n 2´transpositions in Sn is an involution on CPn−2that 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. 1. 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 Sn. In fact, for each n≥3, there is a unique holomorphic map gon CPn−2whose critical set consists of an Snorbit of n 2hyperplanes that gpreserves. This leads to a strong form of critical finiteness that yields several global dynamical results of the type that eluded earlier undertakings. 2000 Mathematics Subject Classification. Primary: 37F45; Secondary: 20C30. Key words. Complex dynamics, equivariant map, reflection group. 128 S. Crass The treatment develops in three stages: (1) Some background on special actions of Snand their associated symmetrical maps. (2) Proofs that the special family of critically-finite maps with Snsymmetry 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. Snacts on CPn−2 The permutation action of the symmetric group Snon Cnpreserves the hyperplane H=(n X k=1 xk= 0)≃Cn−1 and, thereby, restricts to a faithful (n−1)-dimensional irreducible representation. This action on Cn−1projects one-to-one to a group Gn on H:= PH ≃ CPn−2. 2.1. Special orbits and reflection hyperplanes. The smallest Gnorbit consists of the npoints [1 −n, 1,...,1],...,[1,...,1,1−n]. (Square brackets indicate points in projective space.) Corresponding to the n 2transpositions (ij) in Snare n 2involutions xi←→ xj on Hthat generate Gnas a complex reflection group. Each generating involution fixes the point [0,...,0, i z}|{ 1,0,...,0, j z}|{ −1,0,...,0] and pointwise fixes the companion hyperplane {xi=xj}.This pointhyperplane pair gives the only fixed points of the involution. They form Gnorbits of size n 2. For ease of reference, use the term “n 2-hyperplane”. Critically Finite Maps with Symmetry 129 2.2. Coordinates. The transformation A:Cn→Cn−1given by u=Ax, A =     1 0 ... 0−1 0 1 ... 0−1 . . .. . .. . .. . .. . . 0 0 ... 1−1     =aij, aij =     1i=j −1j=n 0 otherwise gives a special system of n−1 coordinates on Hwhere the n-point orbit is [1,0,...,0],...,[0,...,0,1],[1,...,1]. Note that the null space of Ais the euclidean orthogonal complement to H. This change of coordinates has an “inverse” x=Bu, B =       1−n1 1 ... 1 1 1 1 −n1... 1 1 . . .. . .. . .. . .. . .. . . 1 1 1 ... 1 1 −n 1 1 1 ... 1 1        which gives AB =−n In−1, BA =1n−n In where Imis the m×midentity and 1nis the n×nmatrix each entry of which is 1. Accordingly, Aand Binduce isomorphisms between H and CPn−2. In u-coordinates, the n 2-hyperplanes are the n−1 coordinate hyperplanes {uk= 0}and the n−1 2spaces {uk=u`}.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 pkand qk with complementary coordinates. Relative to the uspace, Gnis generated over the permutation action Gn−1of Sn−1on the ukby means of the involution T=       −100... 0 0 −110... 0 0 . . .. . .. . .. . .. . .. . . −100... 1 0 −100... 0 1        that transposes the pair {p1, q1}and fixes the remaining members of the n-point orbit. Note that Tis the uversion of the transformation x1←→ xn. 130 S. Crass Representative points non n−2 hyperplanes Orbit size 2m−1pk= [ k z}| { 1,...,1, n−k−1 z }| { 0,...,0] n−1 k+n−1 k−1=n k qk= [ n−k z}| { 1,...,1, k−1 z }| { 0,...,0] k= 1,...,m−1 2m pk= [ k z}| { 1,...,1, n−k−1 z }| { 0,...,0]   n kk < m 1 2n k=n−1 k−1k=m qk= [ n−k z}| { 1,...,1, k−1 z }| { 0,...,0] k= 1,...,m Table 1. Points determined by intersections of n 2-hyperplanes. 3. Gnequivariants Consider a map f= [f1,...,fn−1] from Hto itself given by homogeneous polynomials in u= (u1,...,un−1) of degree r. In general, fcan be meromorphic; that is, for some p∈Cn−1, f(p) = 0 for every lift of fto Cn−1. We say that fis Gn-equivariant when it sends a group orbit to a group orbit. Algebraically, this means that fcommutes with every element of Gn. Obviously, fis Gn−1-equivariant as well. It readily follows that each component fkis invariant under the stabilizer Zkof uk. Thus, we can express a component by fk= r X `=0 ur−` kAk,` where Ak,` is a degree-` Zkinvariant. Accordingly, each Ak,` is taken to be a polynomial in the elementary symmetric functions in the complementary variables buk= (u1,...,uk−1, uk+1,...,un−1). Alternatively, we can employ the elementary symmetric functions in u when expressing Ak,`. This is a matter of expressing a polynomial in buk Critically Finite Maps with Symmetry 131 in terms of a polynomial in uand a polynomial in bukwith lower degree. Specifically, let b Smand Smbe the degree-melementary symmetric functions in bukand urespectively. Taking S0= 1, the relations b Sm=Sm−ukb Sm−1 give a reductive scheme for the replacement process. An immediate consequence of Gn−1equivariance is that Aj,` =Ak,` := A`for all j, k, `. We can say a bit more concerning the form that Gnequivariants take. First, consider a point athat some element M∈ Gnfixes and observe that Mf(a) = f(Ma) = f(a). Thus, feither sends ato another fixed point of Mor blows up at a —that is, for any lift e fand eaof fand ato Cn−1,e f(ea) = 0. Applying this condition to the n 2-hyperplanes, provided that ais 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 Gnequivariant fsends an n 2-hyperplane to itself. This circumstance forces fkto be divisible by ukand, thereby, requires the terms Arto be a power of Sn−1or to vanish. In particular, when r≤n−1, Ar= 0 so that fk=uk r−1 X `=0 ur−`−1 kA`. By design, the map fhas Gn−1symmetry. To be fully Gn-equivariant, the map must commute with Tas well. This condition places strong restrictions on the A`.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 Gnequivariants with very special properties. 4. Reflection hyperplanes as critical sets: existence, uniqueness, and holomorphy Explicit computation in low-degree cases reveals the existence of a unique holomorphic Gnequivariant whose critical set is precisely the 132 S. Crass n 2-hyperplanes counted with multiplicity two. These maps conform to a general formula. Let g= [g1,...,gn−1] where g`=u3 `G`, G`= n−2 X k=0 (−1)kk+ 1 k+ 3uk `Sn,n−2−k, and Sn,` is the degree-`elementary symmetric function in u1,...,un−1. In the degree-0 case, take Sn,0= 1. By construction, each gis equivariant under the group Gn−1that permutes the u`. In addition, the u3 `factor in each coordinate implies that the maps are doubly critical on n−1 of the n 2-hyperplanes —namely, where u`= 0. Were gto commute with the transformation Tthat generates Gnover Gn−1, symmetry would provide for double criticality on the remaining n−1 2of the n 2-hyperplanes —where uj=uk.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 Gnaction. Theorem 4.3 states that each gis holomorphic on Hwhich implies that gpreserves each n 2-hyperplane Lrather than collapse Lto a lower-dimensional variety; a contraction would force the map to blow up. Thus, gis a family of maps each member of which is holomorphic, doubly-critical on the n 2-hyperplanes, and critically-finite. As a standing assumption, let n≥3. Theorem 4.1. The respective gis T-equivariant, hence, Gn-equivariant. Theorem 4.2. Under the action of Gn,gis 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 gis holomorphic on H. Proof of Theorem 4.1: Propositions 4.5 and 4.8 below establish that g is symmetric under Tas well as under Gn−1. Since Tgenerates Gn over Gn−1,gis Gn-equivariant. The proofs of the propositions rely on a formula that describes how the elementary symmetric functions transform under T. This result Critically Finite Maps with Symmetry 133 was found by pattern detection in low-degree cases. For simplicity of appearance, express the functions Sn,k(u) in the suppressed form Sn,k. Lemma 4.4. For k≤n, the Gn−1invariants Sn,k transform under T according to Sn,k(Tu) = k X `=0 (−1)`n−k+` n−ku` 1Sn,k−`. Proof: Proofs of several technical lemmas appear at [C4]. The argument for the T-equivariance of gexamines the coordinates individually. Proposition 4.5. The factor G1of g1is T-invariant (in a linear as well as projective sense). Proof: The proof amounts to manipulation of sums. Since nis fixed here, let Sk=Sn,k. Consider G1(Tu)= n−2 X k=0 (−1)kk+ 1 k+ 3(−u1)kSn−2−k(Tu)= n−2 X k=0 k+ 1 k+ 3uk 1Sn−2−k(Tu). By Lemma 4.4, G1(Tu) = n−2 X k=0 k+ 1 k+ 3uk 1 n−2−k X `=0 (−1)`n−(n−2−k) + ` n−(n−2−k)u` 1Sn−2−k−`! = n−2 X k=0 k+ 1 k+ 3 n−2−k X `=0 (−1)`k+`+ 2 k+ 2 uk+` 1Sn−2−(k+`)!. Setting m=k+`, G1(Tu) = n−2 X k=0 k+ 1 k+ 3 n−2 X m=k (−1)m−km+ 2 k+ 2 um 1Sn−2−m!. Reversing the order of summation, G1(Tu) = n−2 X m=0 m X k=0 (−1)m−kk+ 1 k+ 3m+ 2 k+ 2 !um 1Sn−2−m = n−2 X m=0 (−1)m(m+ 2)! m X k=0 (−1)kk+ 1 (k+ 3)! 1 (m−k)!!um 1Sn−2−m. 134 S. Crass Lemma 4.6 below gives the sum over k: G1(Tu) = n−2 X m=0 (−1)m(m+ 2)! m+ 1 (m+ 3)!um 1Sn−2−m = n−2 X m=0 (−1)mm+ 1 m+ 3um 1Sn−2−m =G1(u). Lemma 4.6. m X k=0 (−1)kk+ 1 (k+ 3)!(m−k)! =m+ 1 (m+ 3)!. Proof: See [C4]. Corollary 4.7. Each gis T-equivariant in the first coordinate. Proof: Let [·]1specify a map’s first coordinate. Then g1◦T=−u3 1G1◦T=−u3 1G1= [T◦g]1. To establish overall T-equivariance, it suffices to consider the behavior of gunder Tin just the second coordinate. This follows directly from the commutativity of Tand the members τ2,m ∈ Gnthat simply transpose the second and mth basis elements: [0,1,0,...,0] τ2,m ←→ [0,0,...,0,1 |{z} m ,0,...,0] provided that m6= 1, n. Expressed in terms of Sn, this amounts to the commutativity of the disjoint transpositions (1n) and (2m). So, noting that gis Gn−1-equivariant, hence, τ2,m-equivariant, and given that gis T-equivariant in its second coordinate, gm◦T=[g◦T]m =[(τ2,m ◦g◦τ2,m)◦T]m=[τ2,m ◦(g◦T◦τ2,m)]m=[g◦T◦τ2,m]2 =[g◦T]2◦τ2,m = [T◦g]2◦τ2,m = [T◦g◦τ2,m]2 =[τ2,m ◦T◦g]2= [T◦g]m. Proposition 4.8. The second coordinate of gsatisfies the equivariance condition g2◦T= [T◦g]2. Critically Finite Maps with Symmetry 135 Proof: First, express g2◦Tin a way that’s useful for comparison to [T◦g]2. Again, set Sk=Sn,k. Applying Lemma 4.4, g2(Tu)=(u2−u1)3 n−2 X k=0 (−1)kk+ 1 k+ 3(u2−u1)kSn−2−k(Tu) = n−2 X k=0 k+ 1 k+ 3 n−2−k X `=0 (−1)k+` k+2+` k+ 2 u` 1Sn−2−k−`!(u2−u1)k+3. Setting m=k+`, g2(Tu) = n−2 X k=0 k+ 1 k+ 3 n−2 X m=k (−1)mm+ 2 k+ 2 um−k 1Sn−2−m!(u2−u1)k+3. Reversing the order of summation, g2(Tu) = n−2 X m=0 (−1)m m X k=0 k+ 1 k+ 3m+ 2 k+ 2 um−k 1(u2−u1)k+3!Sn−2−m. Lemma 4.9 below establishes a useful identity for the sum over kso that g2(Tu) = u3 2 n−2 X m=0 (−1)mm+ 1 m+ 3um 2Sn−2−m −u3 1 n−2 X m=0 (−1)mm+ 1 m+ 3um 1Sn−2−m −u1u2 n−2 X m=0 (−1)m(um+1 2−um+1 1)Sn−2−m. The first two terms are g2(u) and g1(u) respectively. Since their difference amounts to [Tg(u)]2, Tg(u)2−g2(Tu) = u1u2 n−2 X m=0 (−1)m(um+1 2−um+1 1)Sn−2−m. Adding and subtracting −u1u2Sn−1on the right, Tg(u)2−g2(Tu) = u1u2 −Sn−1+ n−2 X m=0 (−1)mum+1 2Sn−2−m −−Sn−1+ n−2 X m=0 (−1)mum+1 1Sn−2−m!. 142 S. Crass With this, G1(pm) = n−2 X k=0 (−1)kk+ 1 k+ 3Sn,n−2−k(pm) = n−2 X k=n−2−m (−1)kk+ 1 k+ 3m n−2−k. Setting p=n−2−k, G1(pm) = m X p=0 (−1)n−2−pn−p−1 n−p+ 1m p = (−1)n m X p=0 (−1)pn−p−1 n−p+ 1m p. From Lemma 4.13 below, G1(pm) = (−1)n2 (−1)m−1 (n+ 1)n m6= 0. Lemma 4.13. m X p=0 (−1)pn−p−1 n−p+ 1m p=2 (−1)m−1 (n+ 1)n m. Proof: See [C4]. 5. Reflection hyperplanes as critical sets: global dynamics Let Ln−3generically denote an n 2-hyperplane and let Xrefer to the union of the Ln−3. Where mof the Ln−3intersect to form a CPn−2−m, call the resulting space Ln−2−m. (Note that more than mof the Ln−3 can pass through an Ln−2−m.) Not only is gcritically-finite on H≃CPn−2with critical set consisting of the Ln−3hyperplanes, the restriction gLn−2−mis also criticallyfinite, having a collection of the Ln−3−mfor its critical set. In [FS1], such behavior is called strict critical finiteness (Section 7). In fact, all of the Ln−3−mon an Ln−2−mare critical for gLn−2−mthough not with the same multiplicity. Critically Finite Maps with Symmetry 143 5.1. The Fatou set of g.Following standard practice, the Fatou set Fg is where the family of iterates {gk}is normal and the Julia set Jgis the complement of Fg. The behavior of gon an Ln−3plays a central dynamical role. Again, lift gto Cn−1: eg= (g1,...,gn−1) with g`=u3 `G`and G`= n−2 X k=0 (−1)kk+ 1 k+ 3uk `Sn,n−2−k. For a space Lm⊂CPklifted to Ck+1, call the lifted space e Lm+1. Proposition 5.1. For any a∈ Ln−3,gis critical in the direction off of the hyperplane. Proof: By symmetry, consider the e Ln−2given by {u1= 0}. For any a∈ {u1= 0}, the first row of Deg(a) vanishes. Thus, the local behavior of egcollapses points onto e Ln−2. Explicit calculation reveals that the collapse occurs in the direction of (2,1,...,1). Recall that the pmrepresent the point sets of Gnorbits determined by intersecting the Ln−3. Refer to these orbits as “pm-points”. First of all, each such point is superattracting in all directions. Theorem 5.2. Under g, the fixed pm-points are superattracting in every direction. Conversely, the only points that are superattracting in every direction are the pm-points. Proof: To establish that, at pm,gis critical in every direction in CPn−1 show that the Jacobian Degat pmhas rank 1. Here, pmis lifted in the literal way. It then follows that, since eg(pm)6= 0, there are n−2 nonradial directions through pmthat have zero eigenvalue. The Jacobian has the form Deg=(aij) (bij) 0 0  where aij =     3Gi(pm) + ∂Gi ∂ui (pm)i=j ∂Gi ∂uj (pm)i6=j , i, j ≤m bij =∂Gi ∂um+j (pm), i ≤m < j. 144 S. Crass With Sk=Sn,k a straightforward calculation establishes that, for `≤m, ∂Sk ∂u` (pm) = (0k > m m−1 k−1k≤m so that ∂Gi ∂uj(pm) is the same value for i, j ≤mwith i6=j. Similarly, ∂Gi ∂u`(pm) is the same value for ` > m. It remains to show that 3Gi(pm) + ∂Gi ∂ui (pm) = ∂Gj ∂uk (pm) for all i, j, k ≤m. By manipulation of sums, ∂Gi ∂ui (pm) = n−2 X k=0 (−1)n−kn−k−1 n−k+ 1 (n−2−k)Sk(pm) + n−2 X k=0 (−1)n−kn−k−1 n−k+ 1 ∂Sp ∂ui (pm). The second sum is ∂Gj ∂u`(pm) for j, ` ≤mand j6=`. To show that the first sum amounts to −3Gi(pm), notice that, from the proof of Theorem 4.3, n−2 X k=0 (−1)n−kn−k−1 n−k+ 1(n−2−k)Sk(pm) = (−1)n m X k=0 (−1)kn−k−1 n−k+ 2(n−2−k)m k = (−1)n m X k=0 (−1)kn−k−1 n−k+ 1((n−k+ 1) −3)m k = (−1)n m X k=0 (−1)k(n−1−k)m k−3Gi(pm). 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 Critically Finite Maps with Symmetry 145 a binomial expansion of 1 −1. Specifically, m X k=0 (−1)k(n−1−k)m k= (n−1) m X k=0 (−1)km k− m X k=0 (−1)kkm k = (n−1)(1 −1)m+m m X k=1 (−1)km−1 k−1 =m(1 −1)m−1. Thus, the nonzero rows of Deg(pm) are identical and the matrix has rank 1. For the converse claim, consider a point qthat is critical in every direction. When gis restricted to any intersection Lkof hyperplanes each of which is an Ln−2,qis again critical for the restriction gLk. Hence, qlies on some Ln−2that does not contain Lkand so, is determined by the intersection of Ln−2spaces. Now for the issue of the Fatou set Fg. Is there a Fatou component of gthat is not in the basin of a pmpoint? Theorem 5.3. For n= 3,4,Fgconsists of the basins of attraction of the pm-points. Proof: When n= 3, the one-dimensional map ghas three fixed critical pm-points. 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 CP1. 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 CP2has a critical set Csuch that 1) Cis periodic and 2) CP2−C is Kobayashi hyperbolic, then fhas 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,Fgconsists of the basins of attraction of the pm-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. 146 S. Crass The question of whether the basins of the pm-points exhaust Fgcalls for some preparation. Following [U], let Cfbe the critical set of a holomorphic map fon CPm, Df:= ∞ [ k=1 fk(Cf) and Ef:= ∞ \ k=1 fk(Df) be the postcritical set and the ω-limit set of Cfrespectively. Also, the Fatou limit set Λfis where the forward orbits of Fatou components accumulate. In the case of g,Dg=Eg=X. Let p∈Fgand Ube the Fatou component to which pbelongs. For a critically-finite map f, Λf⊂Ef[U, Theorem 5.1]. Accordingly, the forward orbit {gk(p)}of paccumulates on some Ln−3and, by Proposition 5.1, is attracted to that Ln−3—call it Ln−3as well. Accordingly, gn(U)−→ Ln−3. 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 gr(U)∩Ln−36=∅ for some r. An argument might develop in two steps: 1) the orbit of a point that is Fatou for gaccumulates at points that are Fatou for eg:= gLn−3; 2) a point that is Fatou for egis also Fatou for gand, thereby, belongs to a Fatou component in CPn−2. To treat the first claim, let q∈ Ln−3be a limit point of {gk(p)} with gnkK→hwhere h:K→ Ln−3,K⊂Uis a neighborhood of p, and h(p) = q. Suppose that qbelongs to the Julia set Jeg. By Proposition 5.1, gis superattracting at gk(q) in some direction away from Ln−3for all k. This equips qwith a stable set Sq={x|dist(gnk(x), gnk(q)) −→ 0} transverse to Ln−3. If egwere hyperbolic—as in the case n= 4, one might expect that the Kobayashi expansion at qwould produce saddlelike behavior and force Uto contain Julia points for g. Critically Finite Maps with Symmetry 147 To see claim 2) above, let q∈Fegwith a neighborhood e Non which {egk}is normal. Take Nto be the connected neighborhood of qthat is absorbed by e Nand includes e N; that is, Nis the connected component of the stable set of e N Se N=[ x∈ e N Sx where e N⊂Nand Sxis the stable set of x. Every point in Nbelongs to some Sx. Thus, if egnkconverges to ehon e N, then gnkconverges on Nto h(y) = eh(x), y ∈Sx. The claims 1) and 2) imply that some gr(U) intersects Ln−3; indeed, gr(U)∩Ln−3is a Fatou component for eg. By the critical finiteness of eg, the forward orbit of gr(U)∩Ln−3meets some Ln−4in Fatou points for gLn−4. This cascade continues until some gs(U) makes contact with a line L1, in particular, with the Fatou set of gL1. Since gL1has fixed critical points, it has only superattracting basins. The only critical points on L1 are pm-points. Hence, gs(U)∩L1lies in the basin of attraction of some such point. How “large” are the basins of the pm-points? First of all, let Bf:= [ k≥0 f−k(Cf) be the precritical set of f. The following basic result yields that the closure of Bgcontains the Julia set Jg[FS1, Proposition 6.5]. Theorem 5.5. If f:CPk→CPkis holomorphic and CPk−Bfis hyperbolically embedded, Jf⊂Af:= \ n>0[ m>n f−m(Cf). To apply this result to g, we must see that it satisfies the hypotheses. By Theorem 4.3, gis holomorphic on CPn−2. Two theorems of M. Green imply that Ln−1−m−BgLn−1−m is hyperbolically embedded in Ln−1−m(taking Ln−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−→ Ln−1−m−BgLn−1−m is holomorphic. Then φ(C) omits at least n−m+ 1 hypersurfaces in Ln−1−m, namely, some Ln−m−2spaces and their preimages. By one 148 S. Crass 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 Cg⊂g−1(Cg), Jg⊂Ag=Bg. We can now establish a bit of Fg’s global structure. Theorem 5.6. Under the assumption that Conjecture 5.4 holds, the Fatou set Fgis dense in H. Proof: Consider j0∈Jgand let U0be a neighborhood of j0. By Theorem 5.5, some precritical points meet U0so that, for some m, gm(U0)∩Cg6=∅. If U1:= gm(U0)∩Ln−3 fails to contain Julia points, the case is made. Otherwise, take a Julia point j1∈U1, a neighborhood of j1. The map gLn−3is critically finite with critical set Cn−3in the intersection of Ln−3and the hyperplanes in Xdifferent from Ln−3. Hence, Cn−3is a collection of Ln−4spaces. Implementing the argument given for j0and U0under gusing j1and U1under gLn−3produces a neighborhood of a Julia point j2on some Ln−4. The descent continues until it reaches a Julia point jn−3and neighborhood Un−3on an L1. Thus, Un−3meets the Fatou set of gL1. Since gL1has fixed critical points that are pm-points, its Fatou set consists of the superattracting basins of those pm-points. Accordingly, Un−3—hence, U0— contains points in Fg. 5.2. A query on the structure of g’s Julia set. For the restricted map bg=gLn−3, the Julia set is given by Jbg=Jg∩Ln−3. The inclusion Jbg⊂Jg∩ Ln−3is clear. If x /∈Jbg, then xbelongs to a basin of a pm-point so that x /∈Jg. At each point p∈Jbg, the map is superattracting in the direction away from Ln−3. Thus, there is a “stable set” Spof points in Jgwhose orbits are attracted to the orbit of p. Accordingly, there is a stable bundle over Jbg SJbg:= [ p∈Jbg Sp⊂Jg. Critically Finite Maps with Symmetry 149 Are the Spone-dimensional manifolds? Are the preimages of the SJbg dense in Jg? In the case n= 4, grestricts to a critically-finite map bgon an L1that is one of the six lines of reflection for G4. Figure 4 displays the three basins of attraction for bg. The Julia set Jbgconsists of the boundaries of these basins. For each Julia point p∈ L1, there is an Spaway from the line. What can be said about the structure of SJbg? What about the points K:= Jg−[ X SJbg that are not absorbed by X? On an L1, each Julia point is non-wandering and has a contracting direction onto L1and an expanding direction in L1. For a hyperbolic map on CP2, the literature describes a grading of the non-wandering set Ω by the expanding dimension [FS2]: Ω = Ω0∪Ω1∪Ω2. The pm-points comprise Ω0and ∪XJbg⊂Ω1. The non-wandering points not on Xbelong to K. Since any neighborhood of such a point pcontains an open set that is attracted to X, there is expansion at p. Does it happen that Ω∩K⊂Ω2 so that gis hyperbolic? 6. Geometry and dynamics in low-dimension To avoid confusion, let gn+1 represent the particular map gon the respective Gn-symmetric H. 6.1. The one-dimensional case: g4and Halley’s method. When n= 3, the reflecting “hyperplanes” consist of a three-point orbit. With these points located at {1, ρ, ρ2|ρ=e2π i/3}, the map’s inhomogeneous expression on {u26= 0}is z−→ z(z3−2) 2z3−1. We can realize the G3action on CP1by 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 150 S. Crass boundary between hemispheres and the 3-points {1, ρ, ρ2}are vertices where four faces congregate. Consider the degree-4 map that fixes the vertices of each face and sends one face Fto four others: Fitself and the three faces in the hemisphere not containing F. This symmetrical construction results in G3-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 g4. Since g4has periodic critical points, the superattracting basins constitute its Fatou set and, moreover, have full measure in CP1. A portrait of the basins appears in Figure 1. It turns out that g4is Halley’s Method —a variation on Newton’s Method— for 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 z3−1. Figure 1. Dynamics of g4on the S3-symmetric CP1 Critically Finite Maps with Symmetry 151 6.2. The map in two dimensions. Since gn+1 has real coefficients, it preserves the RPn−2of points whose coordinates can be expressed by real numbers. Call this space R. Under G4,Rhas 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. Figure 2 displays the basins of attraction of g5on R. In the affine plane of the picture, the vertices of the cube are (0,0),(1,0), −1 2,±√3 2! while the three face-centers are the edge-midpoints −1 2,0, 1 4,±√3 4! of the equilateral triangle formed by the three vertices that are not (0,0). The map is given by (x, y)−→ 3`15 x4+ 12 x5−30 x2y2−5y4+ 20 x y4´ 1−10 x2+ 20 x3+ 30 x4+ 40 x5−10 y2−60 x y2+ 60 x2y2−80 x3y2+ 30 y4−120 x y4, 24 `−5x y3+ 5 x2y3+y5´ 1−10 x2+ 20 x3+ 30 x4+ 40 x5−10 y2−60 x y2+ 60 x2y2−80 x3y2+ 30 y4−120 x y4!. 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 G4. Viewing the “hemi-cube” 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. Returning to ucoordinates, one of the six mirrors —say, {u3= 0}— is Z2-stable. Restricted to this line, g5has three superattracting points: •A two-point Z2orbit of type p1points [1,0,0] and [0,1,0] (where {u2= 0}and {u1= 0}intersect {u3= 0}). •A one-point Z2orbit of the point p2= [1,1,0] (where {u1=u2} intersects {u3= 0}).