CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS

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1 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS LAURA DEMARCO AND KEVIN PILGRIM Abstract. Let M be the mouli space of one-imensional, egree 2, complex polynomial ynamical systems. The escape rates of the critical points etermine a critical heights map G : M R 1. For generic values of G, we show that each connecte component of a fiber of G is the eformation space for twist eformations on the basin of infinity. We analyze the quotient space T obtaine by collapsing each connecte component of a fiber of G to a point. The space T is a parameterspace analog of the polynomial tree T (f) associate to a polynomial f : C C, stuie in [6], an there is a natural projection from T to the space of trees T. We show that the projectivization PT is compact an contractible; further, the shift locus in PT has a canonical locally finite simplicial structure. The topimensional simplices are in one-to-one corresponence with topological conjugacy classes of structurally stable polynomials in the shift locus. 1. Introuction This article continues the stuy, initiate by Branner an Hubbar in [3, 4], of the global structure of the mouli space of polynomials f : C C etermine by the ynamics on the basin of infinity, X(f) = {z C : f n (z) as n }. Associate to a pair (f, X(f)) are analytic invariants, such as the uniformize Böttcher coorinates of its critical points, an combinatorial invariants, such as the tree T (f) efine in [6, 13] or the tableau of [4]. In this article an its sequel [9], we investigate the extent to which such invariants etermine the ynamics of the polynomial f on its basin of infinity. This stuy is partly motivate by a esire to unerstan the shift locus in the mouli space, the set of polynomials f where all critical points lie in X(f). In the shift locus, the conformal conjugacy class of a polynomial is uniquely etermine by its restriction to the basin of infinity, an all such polynomials are topologically conjugate on their Julia sets. The shift locus has complicate topology, which can be seen in the rich variation of the global ynamics of these polynomials [2], an there is currently no known invariant which classifies its topological conjugacy classes. In a sequel, we provie a combinatorial metho to complete the classification of polynomial Date: July 16, 2010.

2 2 LAURA DEMARCO AND KEVIN PILGRIM ynamical systems restricte to their basins of infinity. Here we concentrate on the general topological features of conjugacy classes an how they fit together in the mouli space Critical heights. Suppose f is a complex polynomial of egree 2. Its escape-rate function is given by 1 G f (z) = lim n n log+ f n (z). Let {c 1,..., c 1 } be the multiset of critical points of f, each repeate accoring to its multiplicity, an labele so that G f (c 1 )... G f (c 1 ). Finally, let The map H = {(h 1,..., h 1 ) : h 1 h 2... h 1 0}. escens to the critical heights map f (G f (c 1 ),..., G f (c 1 )) G : M H on the space M of complex affine conjugacy classes of polynomials of egree. We begin with the following observation: Theorem 1.1. The critical heights map G : M H is continuous, proper an surjective. The continuity an properness of G follow easily from [3]. Surjectivity follows from basic properties of analytic maps, since the lift of G to the space of maps with marke critical points is pluriharmonic on the locus where all critical points have positive heights. Details are given in 2. We stuy the ecomposition of M into connecte components of the fibers of G. This critical-heights ecomposition is relate to twisting eformations on the basin of infinity: Theorem 1.2. For generic values of G : M H, the fiber of G is a finite, isjoint union of smooth real ( 1)-imensional tori; each connecte component coincies with a twist-eformation orbit. In particular, the connecte components of a generic fiber are precisely the topological conjugacy classes of polynomials within that fiber. The generic values of Theorem 1.2 have a precise characterization: we show that a value (h 1,..., h 1 ) of G satisfies the conclusion of Theorem 1.2 if an only if h i > 0 for all i an h i n h j for all i j an n Z. Dynamically, these generic critical heights correspon to polynomials in the shift locus with all critical points in istinct foliate equivalence classes. The twist eformation, introuce in [17], is a quasiconformal eformation of the basin of infinity of a polynomial which twists its funamental annuli an preserves its critical heights; the turning curves of [3] are a special case. See 5.

3 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS Monotone-light factorization. A close, proper map between Hausorff spaces factors canonically as a monotone map (i.e. one whose fibers are connecte) followe by a light map (one whose fibers are totally isconnecte); see [5, Theorem I.4.3]. Below, we apply this to the critical heights map. The critical heights map factors as a composition of continuous, proper, an surjective maps M B T H. Here, B is the space of conformal conjugacy classes (f, X(f)) of restrictions of polynomials to their basins of infinity, equippe with the Gromov-Hausorff topology, introuce in [7]; T is the space of polynomial metric trees with self-maps (F, T ), also equippe with the Gromov-Hausorff topology, introuce in [6]. We remark that the space T iffers somewhat from its original efinition, in that we have inclue the unique trivial tree associate to polynomials with connecte Julia set. See 4 for etails. The main result of [7] states that the projection M B is monotone. On the other han, the critical heights map on the space of trees T H is light (Lemma 4.1). The fibers of the projection B T can be isconnecte, an the reason is not too surprising. In forming the tree T (f) of a polynomial f, connecte components of the level sets of G f are collapse to points, thus forgetting the complicate planar graph structure of the singular components. We give explicit examples in the sequel [9], but we remark here that even for generic height values in egree = 3, there are istinct topological conjugacy classes of polynomials with the same tree. We efine T to be the quotient space of M obtaine by collapsing connecte components of the fibers of G to points; it is a parameter-space analog of the tree for a single polynomial. The facts that T H is light an M B is monotone imply that we obtain a new sequence of continuous, proper, surjective maps (1.1) M B T T H through which the critical heights map G must factor. By construction, the quotient space T is the unique monotone-light factor for G. That is, the fibers of M T are connecte (an, in fact, always contain a non-egenerate continuum), while the fibers of T H are totally isconnecte. Moreover: Theorem 1.3. The fibers of the projection T T are totally isconnecte; an the fibers are finite over the shift locus in T. Furthermore, for each h 0, the fiber over the unique tree with uniform critical heights (h, h,..., h) is a single point. From the efinitions we euce: is the unique monotone-light factor for the tree projec- Corollary 1.4. The space T tion map M T.

4 4 LAURA DEMARCO AND KEVIN PILGRIM T The objects of the space T are at present somewhat mysterious; the notation has been chosen because elements shoul represent polynomial trees, augmente by a certain amount of combinatorial information. The extra information neee to specify an element of T from the corresponing element of T is the subject of [9]. In particular, we provie examples there showing that the carinality of the fibers of T T is uniformly boune if an only if = 2 (in which case it is a bijection). Theorem 1.2 shows that T is generically the orbit space for the twisting eformation; in [8] we showe that the full space T can be interprete as the hausorffization of the twist-orbit space Cone structures an stretching. Stretching is a quasiconformal eformation of a polynomial, introuce in [3], which expans (or contracts) the basin of infinity in the graient irection of its escape-rate function. While not continuous in the polynomial, stretching nevertheless efines a continuous action of R + on each of the quotient spaces in (1.1), which is free an proper on the complement of the connecteness locus (Lemma 5.1). The R + -actions are equivariant with respect to the projection maps in (1.1). The critical heights map G satisfies G(s [f]) = s G([f]) for all s R +. The connecteness locus of each quotient space is the unique fixe point of the stretching action, an it contains the set of accumulation points of each orbit as s ecreases to zero. We obtain: Theorem 1.5. The stretching operation inuces a cone structure on each of the spaces B, T, T, an H. Specifically, they are cones over the compact sets B (1), T (1), T (1), an H (1), consisting of points with maximal critical escape rate equal to 1, with origin at the quotient of the connecteness locus. Because of the cone structure, there is a natural projectivization of each quotient space of M in (1.1), ientifie with the slice with maximal critical height 1. For the space of trees T, this projectivization is the space PT of [6]. In [7], we introuce a eformation of polynomials which flows critical points upwar along external rays until they reach the height of the fastest escaping critical point. While this cannot be canonically efine on M (1) or B (1) (even in the shift locus, ue to the collision of external rays), the ambiguity is avoie when passing to the quotient space T. Using this upwar flow, we euce: Theorem 1.6. The projectivization PT is compact an contractible. The corresponing statement for the projectivize space of trees PT was prove in [6].

5 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS 5 PST 3 PST 3 PSH 3 1 1/3 1/3 2 1/3 3 1/3 4 1/3 5 Figure 1.1. The simplicial projections of Theorem 1.7 in egree = 3, rawn for critical heights 1/3 5 h 2 /h 1 1 (not to scale). The eges of PST3 correspon to structurally stable topological conjugacy classes of polynomials in the shift locus of M 3. Note that PST3 has one vertex more than PST 3 at height 1/ Stratification of the shift locus. The shift locus in each of the spaces of (1.1) is the subset of points with all critical heights positive. It forms a ense open subset of each of these quotient spaces of M. Restricting to the shift locus yiels a sequence of surjective proper maps (1.2) SM SB ST ST SH. The first map is a homeomorphism [7, Theorem 1.1], an SH is the subset of H with all coorinates positive. Each of the spaces in (1.2) is stratifie by the maximal number of inepenent critical heights, where positive real numbers x an y are inepenent in egree if there is no integer n such that x = n y. Dynamically, the N-th stratum SM N consists of polynomials with exactly N istinct critical foliate equivalence classes. By the quasiconformal eformation theory of [17], the connecte components of the open stratum SM 1 are precisely the topological conjugacy classes of structurally stable polynomials in the shift locus. We enote the N-th stratum on each of the spaces in (1.2) with the superscript N. The strata ST N, ST N, an SHN are non-connecte real manifols, with each component homeomorphic to R N 1. The maps in (1.2) preserve the strata, an the strata are invariant uner stretching. Passing to the quotients by stretching, which we enote with the prefix P for projectivization, we obtain a new sequence of surjective proper maps (1.3) PSM PSB PST PST PSH. It follows that the stratifications escen to the projectivize shift loci, an that the maps in (1.3) also preserve the corresponing strata.

6 6 LAURA DEMARCO AND KEVIN PILGRIM Theorem 1.7. The spaces PST, PST, an PSH carry a canonical, locally finite simplicial structure, an the projections PST PST PSH are simplicial. By the above-mentione escription of structurally stable maps in the shift locus from [17], we obtain: Theorem 1.8. The set of globally structurally stable topological conjugacy classes of polynomials in the shift locus of M is in bijective corresponence with the set of top-imensional open simplices in PST. It woul be useful, therefore, to unerstan the projectivize shift locus better. A classification of the stable conjugacy classes in the shift locus is one of the goals of the article [9]. In egree = 2, the projectivization PST2 is a point, while for egree = 3 it is an infinite simplicial tree. See Figure 1.1. Using the combinatorics of [9], a computational enumeration of the eges of the tree PST3 is carrie out in [10]. 2. Polynomials In this section we efine the mouli space M an present some basic properties of the critical heights map G : M H. We prove Theorem 1.1, showing that G is surjective Monic an centere polynomials. Every polynomial f(z) = a 0 z + a 1 z a, with a i C an a 0 0, is conjugate by an affine transformation A(z) = az + b to a polynomial which is monic (a 0 = 1) an centere (a 1 = 0). A monic an centere representative is not unique, as the space of such polynomials is invariant uner conjugation by A(z) = ζz where ζ 1 = 1. In this way, we obtain a finite branche covering P M from the space P C 1 of monic an centere polynomials to the mouli space M of conformal conjugacy classes of polynomials. Thus, M has the structure of a complex orbifol of imension 1. It is sometimes convenient to work in a space with marke critical points. Let H C 1 enote the hyperplane given by {c = (c 1,..., c 1 ) : c c 1 = 0}. Then the map ρ : H C P given by (2.1) ρ(c; a)(z) = z 0 1 (ζ c i ) ζ + a i=1

7 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS 7 gives a proper polynomial parameterization of P by the location of the critical points an the image of the origin. Setting P = H C, we refer to P as the space of critically marke polynomials. The marke shift locus is the subset of P efine by where f = ρ(c; a) of equation (2.1). S = {(c ; a) P : G f(c i ) > 0 for all i} 2.2. Critical escape rates. The escape rate function G f (z) = lim n 1 n log + f n (z) is continuous in (f, z) P C. The maximal escape rate M(f) = max{g f (c) : f (c) = 0} is a continuous, proper function on P ; see [3]. The efinition implies that the escape rate satisfies the functional equation G f (f n (z)) = n G f (z) for all n N. On the open subset of P C where G f (z) > 0, the function (f, z) G f (z) is a locally uniform limit of pluriharmonic functions n log f n (z). So the map (f, z) G f (z) is pluriharmonic where G f (z) > 0. Since G A f A 1 = G f A 1 for all complex affine maps A, the critical heights map inuce by sening G : M H f (G f (c 1 ),..., G f (c 1 )) is well efine; recall that we liste critical points accoring to their multiplicity, an labele them so that G f (c 1 )... G f (c 1 ). The critical heights map G is proper since it is a factor of the proper map f M(f). (We remark that the choice of labeling for the critical points is somewhat artificial; we coul just as well have efine G as a map to the space of unorere ( 1)-tuples. The ecomposition of M forme by the istinct connecte components of fibers of G within M woul be the same. ) The critical heights map G lifts to a map G : P [0, ) 1 on the space critically marke polynomials by setting G (c ; a) = (G f (c 1 ),..., G f (c 1 )), where f = ρ(c; a) is given by equation (2.1) an c = (c 1,..., c 1 ). Since f = ρ(c; a) is continuous in c an a an G f (z) is continuous in f an z, the map G is continuous. It is proper since the iagram P G [0, ) 1 ρ P G commutes an the left-han vertical an bottom maps are proper. We conclue: H

8 8 LAURA DEMARCO AND KEVIN PILGRIM Lemma 2.1. The restriction of the critical heights map to the marke shift locus is pluriharmonic an proper. G : S (0, ) Proof of Theorem 1.1. It remains only to show surjectivity of G : M H. Referring to the commutative iagram above, it is enough to show that the lifte map G : P [0, ) 1 is surjective. Since it is proper, in orer to show that it is surjective, it will suffice to show that the restriction to the marke shift locus G : S (0, ) 1 is both open an close. That it is close is a consequence of its properness, prove in Lemma 2.1 above. To prove that it is open, we work with holomorphic maps, an appeal again to Lemma 2.1. Given f P, the Böttcher map w = ϕ f (z) is the unique analytic isomorphism ϕ f : {z : G f (z) > M(f)} {w : w > e M(f) } tangent to the ientity near infinity an satisfying ϕ f f ϕ 1 f (z) = z. The map (f, z) ϕ f (z) is analytic in f an z (see e.g. [4, Prop 3.7]). Fix an integer n 1. Let S (n) = {(c; a) S : G f(f n (c i )) > M(f) for all 1 i 1} where f = ρ(c; a). That is, S (n) is the set of critically marke polynomials f for which the nth iterates of the critical points of f lie in the omain of the Böttcher map ϕ f. By [7, Lemmas 5.1, 5.2], the set S (n) is connecte. Let W (n) C 1 be the set of ( 1)-tuples (w 1,..., w 1 ) such that w i > 1 an n log w i > max log w j 1 j 1 for each i = 1,..., 1. Let H (n) (0, ) 1 be the set of ( 1)-tuples (h 1,..., h 1 ) such that n h i > for each i = 1,..., 1. Since H max 1 j 1 h j (n) = 1 i,j=1 {(h 1,..., h 1 ) : n h i > h j }, the set H (n) is convex, hence connecte. The functional equation satisfie by the escape rate implies that the function Φ n : S (n) W (n) given by Φ n (c ; a) = (ϕ f (f n (c 1 )),..., ϕ f (f n (c 1 ))) inee takes values in W (n). Restricte to S (n), the map G is a composition of maps of omains S Φn (n) W Λn (n) H (n)

9 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS 9 where Λ n (w) = (log w 1,..., log w 1 ) is an open map. The map Φ n is analytic map between omains in C 1. It is also proper since it is a factor of the proper map G ; see Lemma 2.1. By [20, Theorems , ], the map Φ n is open; we remark that it is also surjective, an that the set of regular values is open. Since the omains S (n), n = 1, 2, 3,... exhaust S, we conclue that G : S (0, ) 1 is open Critical heights in egree 2. Finally, we point out that the critical heights map is a well-known object in egree = 2 where there is a unique critical point. The mouli space M 2 C is parametrize by the polynomials f c (z) = z 2 + c, c C an the critical heights map G : C H 2 = [0, ) is given by G(c) = G c (0). It is known to be equal to 1/2 times the Green s function for the Manelbrot set C 2. As the Manelbrot set is connecte, all fibers of G are connecte. The fibers over positive heights are the equipotential curves, thus each is a simple close loop aroun C 2. See [11]. 3. The quotient space T Recall that T is the quotient space of M obtaine by collapsing each connecte component of a fiber of G : M H to a point. In this section, we escribe some basic topological properties of T, an we see that there are only finitely many points in each shift locus fiber Degree 2. As escribe in 2.4, the critical heights map in egree 2 is simply (a multiple of) the Green s function for the Manelbrot set C 2 in M 2, so all fibers are connecte. It is immeiate to see that the critical heights map inuces a homeomorphism T 2 H 2 = [0, ). Thus in egree 2, the space T 2 coincies with the space of trees T Monotone-light factor. Proposition 3.1. The quotient space T is Hausorff, separable, an metrizable, an the projection M T is proper an close. Proof. Properness follows since the map M T is a factor of the proper map M H. From [5, Thm. I.3.5], the ecomposition of M into the fibers of G is upper semicontinuous, an the quotient space obtaine by collapsing fibers of G to points is homeomorphic to the image H. By [5, Prop. I.4.2], the ecomposition of M into the connecte components of the fiber of G is also upper semicontinuous.

10 10 LAURA DEMARCO AND KEVIN PILGRIM By [5, Prop. I.1.1, I.2.1, I.2.2], the quotient space T obtaine by collapsing these components to points is Hausorff, separable, an metrizable, an the projection map is close Finiteness of fibers. By the efinition of T, the map T H has totally isconnecte fibers. We observe here that these fibers are finite when all critical heights are positive. There is, however, no uniform boune on their size in any egree 3 (see Corollary 4.3). Lemma 3.2. Fibers of G in the shift locus of M have finitely many connecte components. Proof. Recall that we can lift G to a pluriharmonic map G : S (0, ) 1 on the critically marke shift locus, as escribe in 2.2. The fibers of G in S are locally connecte [1] an compact (by properness), thus have only finitely many connecte components. It is well known that the connecteness locus C in M is connecte. See [11] for a proof in egree 2, [3] for egree 3, an [16] for a proof that C is cell-like in every egree. Thus, there is a unique point of T with all critical heights equal to 0. More generally, we have: Lemma 3.3. For any height h 0, the locus of polynomial classes in M with all critical heights equal to h is connecte. Thus, there is a unique point in T with all critical heights equal to h. Proof. For h = 0, the locus is simply the connecteness locus C, which is connecte. For h > 0, we argue as follows. Given a polynomial f, let S(f, h) M be the subset represente by polynomials g such that (i) there is a holomorphic isomorphism {G f > h} {G g > h} conjugating f to g, an (ii) G g (c) h for all critical points c of g. The set S(f, h) epens only on the conjugacy class of f in M. We prove in [7] that for all polynomials f an all h > 0, the set S(f, h) is connecte. In particular, for any f with maximal escape rate M(f) h, this set S(f, h) is precisely the fiber G 1 (h, h,..., h) in M. Proofs for h > 0 were also given in [14] an [12]. 4. Trees In this section, we remin the reaer about trees an the space of trees. We prove that the critical heights map T H has totally isconnecte fibers, which implies that the projection T T is well efine. We also observe that the fibers of T H have unboune carinality for all 3, showing that the critical heights map T H also has unboune egree, even in the shift locus. Finally, we prove Theorem 1.3 about the projection T T.

11 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS The tree of a polynomial. Fix a polynomial f, an let G f : C [0, ) enote the escape rate, as efine in the Introuction. The tree associate to f is the monotone-light factor of G f : the quotient C T (f) is obtaine by collapsing each connecte component of a level set of G f to a point. There is a natural simplicial structure on the subset of T (f) which is the quotient of the basin of infinity X(f), realizing this open subset as an infinite locally-finite simplicial tree. The ynamics of f inuces a map F : T (f) T (f) which preserves the natural simplicial structure The space of trees. We begin by recalling the topology on the space T of egree polynomial trees. Fix a polynomial f an let (F, T (f)) be its associate tree equippe with ynamics. The escape-rate function G f of f inuces a height function h : T (f) [0, ) an a metric on T (f) so that the istance between ajacent vertices is their ifference in height. Trees (F 1, T 1 ) an (F 2, T 2 ) are equivalent if there is an isometry i : T 1 T 2 which conjugates F 1 to F 2. There is then a unique tree (F 0, T 0 ) associate to all polynomials with connecte Julia set; we call this the trivial tree. Let T enote the set of equivalence classes of such pairs (F, T (f)). The topology on T can be efine as a Gromov-Hausorff topology, just as on B. The maximal escape rate M(f) of a polynomial is exactly the height of the highest branch point in T (f). The level l Z of a vertex v in T (f) is the number of iterates n so that M(f) h(f n (v)) < M(f). Roughly speaking, trees are close in the topology on T if the truncate ynamical systems (F N, T N ) are close for some large N, where T N is the subtree of vertices of levels l < N. Specifically, a basis of open sets can be efine in terms of two parameters ε > 0 an N N. An open neighborhoo of a nontrivial tree (F, T ) consists of all trees (F, T ) for which there is an ε-conjugacy of the restricte trees (F N, T N ). Open neighborhoos of the trivial tree (F 0, T 0 ) are all trees with maximal escape rate < ε. With these efinitions in place, the proofs of [6] can be use to show that the projection M T is continuous an proper. See [6, Theorem 1.4], where the projection was only efine on M \ C Critical heights on trees. Lemma 4.1. The fibers of the critical heights map T H are totally isconnecte; the fibers are finite in the shift locus. Proof. Suppose a fiber U of the critical heights map T H contains at least two points (F, T ) (F, T ). There must be a level N at which the truncate trees (F N, T N ), (F N, T N ) fail to coincie. There are only finitely many possibilities for these truncate trees. So U is a isjoint union of relatively open subsets where the truncate tree is constant. Therefore U is totally isconnecte. The secon statement follows immeiately from Lemma 3.2 an the fact that the critical heights map M H factors through the space of trees: M T H.

12 12 LAURA DEMARCO AND KEVIN PILGRIM One can also see this irectly: by [6, Theorem 5.7] a tree in the shift locus is uniquely etermine by the subtree own to the level of its lowest critical point. There are only finitely many combinatorial options once the critical heights are fixe Critical heights an the realization of trees. The tree-realization theorem [6, Theorem 1.2] asserts every abstract metric polynomial tree (F, T ) arises from some polynomial. Using this, one can give an alternative proof of Theorem 1.1. Inee, using the axioms for polynomial trees, it is easy to construct an abstract tree (F, T ) with any given collection of heights in H. The simplest construction woul place all critical points along a line which joins the Julia set J(F ) with an which is fixe by F, builing the ynamical tree inuctively as the height escens. Because G : M H factors through the space of trees T, we see immeiately that G is surjective. As another application of tree realization, we fin: Lemma 4.2. The carinality of the fibers of the critical heights map T H is unboune for all 3. Proof. Suppose a tree of egree 3 has exactly two critical vertices v an v, with heights h(v) > h(v ) > 0, with v of multiplicity 1 an v of multiplicity ( 2). There are then 3 eges ajacent to v, two below an one above. The two eges below v are istinguishe, as one has egree 1 an the other has egree 1 > 1. If v takes > N iterates to reach the height of v, then there are at least 2 N choices for its itinerary on the eges below an ajacent to v. All such itineraries are realizable as trees, using the axioms for abstract polynomial trees, an realizable as polynomials, by the realization theorem of [6]. Corollary 4.3. The carinality of the fibers of the critical heights map T unboune for all 3. H is H factors through T H, so the statement is imme- Proof. The projection T iate from Lemma Proof of Theorem 1.3. We first prove that the the map T T is well efine an that its fibers are totally isconnecte. Consier the iagram mon M T mon M / light light T H light

13 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS 13 where the mile space is the canonical monotone factor of M T. Lemma 4.1 implies that the bottom map is light, so the uniqueness of monotone-light factorizations yiels that the spaces M / an T are the same, i.e. that the map inicate by the ashe arrow is the ientity. From Lemma 3.3, we know that the fiber of G over heights (h, h,..., h) is connecte. Therefore, the fiber of T T over trees with uniform critical heights is a point. 5. Quasiconformal eformations In this section, we iscuss quasiconformal eformations of polynomials supporte on the basin of infinity. We prove Theorems 1.2 an 1.5. We begin with a continuity statement that we will use several times Branner-Hubbar motion. The upper half-plane H = {τ = t + is : s > 0} may be ientifie with the subgroup of GL 2 R consisting of matrices ( ) 1 t 0 s with t R an s > 0, regare as real linear maps τ of the complex plane to itself via τ (x + iy) = (x + ty) + i(sy). Note that the parabolic one-parameter subgroup {s = 1} acts by horizontal shears, while the hyperbolic subgroup {t = 0} acts by vertical stretches. The Branner-Hubbar wringing motion of [3] is an action H M M. Explicitly, for each polynomial f with isconnecte Julia set, we consier the holomorphic 1- form ω f = 2 G f on the basin of infinity X(f). In the natural Eucliean coorinates of (X(f), ω f ), the funamental annulus A(f) = {M(f) < G f (z) < M(f)} may be viewe as a rectangle in the plane, of with 2π an height ( 1)M(f), with vertical eges ientifie. The wringing action is by the linear transformation τ on this rectangle, transporte throughout X(f) by the ynamics of f. For polynomials in the connecteness locus, where M(f) = 0, the action is trivial. More precisely: suppose f P an let µ f be the f-invariant Beltrami ifferential given by ω f ω f on X(f) an by 0 elsewhere. By the Measurable Riemann Mapping Theorem, for each τ H, there is a unique quasiconformal homeomorphism ϕ τ : C C tangent to the ientity at infinity an satisfying ϕ τ = iτ 1 ϕ τ iτ + 1 µ. If we then set f τ = ϕ τ f ϕ 1 τ, the map f τ is again an element of P. The escape-rate function of f τ satisfies (5.1) G fτ (ϕ τ (z)) = s G f (z).

14 14 LAURA DEMARCO AND KEVIN PILGRIM where s = Im(τ). The map τ f τ is analytic. Composing with the projection to M, we obtain a map H M obtaine by wringing f. Suppose now g = A f A 1. Then ω f = A ω g, hence µ f = A µ g. It follows that g τ = A f τ A 1 an hence that wringing gives a well-efine action H M M whose orbits are analytic. The action of the above parabolic subgroup is known as turning; it preserves critical heights. The action of the hyperbolic subgroup is calle stretching. Stretching is known to be iscontinuous on M ue to the phenomenon of parabolic implosion [19, Cor. 3.1]; see also [21], [15]. Nevertheless we have Lemma 5.1. The wringing motion escens to a continuous action on each of the spaces B, T, T, an H. In each of these spaces, the point corresponing to the connecteness locus is a global fixe point. Away from this fixe point the action by stretching is free an proper. The turning action is nontrivial on B but trivial on T, T, an H. Proof. We begin by verifying that wringing escens continuously to the space B. Suppose f n, f are polynomials representing elements of M, we fix τ n τ in H, an assume that basins (f n, X(f n )) (f, X(f)) in the Gromov-Hausorff topology on B. Since the forgetful map M B is proper, we may assume f n f for concreteness. Let µ n be the f n -invariant Beltrami ifferentials on C corresponing to the wringing of X(f n ) by τ n ; they are supporte on X(f n ). Similarly, let µ correspon to τ; it is f-invariant an supporte on X(f). Suitably normalize, there are unique quasiconformal maps h n : C C an h : C C whose complex ilations are almost everywhere µ n an µ, respectively [3, Prop. 6.1]. Let g n = h n f n h 1 n an g = h f h 1. By construction, the maps h n are uniformly quasiconformal, so we may pass to a subsequence so that h n h an g n g. It follows that h conjugates f to g, while h conjugates f to g. Let µ be the complex ilatation of h. By [3, Lemma 10.1], µ = µ on X(f). It follows g an g are holomorphically conjugate on their basins of infinity, thus g = g in B. So wringing is continuous on B. By equation (5.1), wringing scales the critical height vector associate to each polynomial. It follows that on B, wringing preserves the fibers of B T an so escens continuously to an action on T. The remaining assertions follow easily Twisting an the Teichmüller space. Given a polynomial, there is a canonical space of marke quasiconformal eformations supporte on the basin of infinity. The general theory, evelope in [17], shows that this space amits the following escription.

15 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS 15 Fix a polynomial f P. The foliate equivalence class of a point z in the basin X(f) is the closure of its gran orbit {w X(f) : n, m Z, f n (w) = f m (z)} in X(f). Let N be the number of istinct foliate equivalence classes containing critical points of f. These critical foliate equivalence classes subivie the funamental annulus A(f) into N funamental subannuli A 1,..., A N linearly orere by increasing height. It turns out one can efine wringing motions via affine maps on each of the subannuli A j inepenently so that the resulting eformation of the basin X(f) is continuous an well efine. The eformations of each subannulus are parameterize by H, so as in 5.1 we obtain an analytic map H N M. By varying the map f as well, we get similarly an action H N M N M N where now M N is the locus of maps with exactly N critical foliate equivalence classes. The wringing by τ = t + is H applie to f M N is the action of vector ( ) 2π m1 t ( 1)M(f) + is,..., 2π m N t ( 1)M(f) + is H N where m j is the moulus of the j-th subannulus of A(f), so that m j = ( 1)M(f)/2π. j The action of R N by the parabolic subgroup in each factor is calle twisting. By construction, the twisting eformations preserve critical heights. Let S N S be the locus represente by polynomials in the shift locus with exactly N istinct critical foliate equivalence classes. Lemma 5.2. The wringing motion on each funamental subannulus efines a continuous action H N S N S N. For each f, the orbit map H N {[f]} S N lattice of translations in R N. is analytic, an the stabilizer of [f] is a Proof. Continuity follows from the same arguments given in the proof of Lemma 5.1; analyticity follows from the analytic epenence of the solution of the Beltrami equation. To prove the secon assertion, suppose f represents an element of S N. Let A j be the jth funamental subannulus of f. Since f belongs to the shift locus, j = lcm{eg(f n : Ãj A j )} < where the least common multiple is taken over all connecte components of all iterate inverse images Ãj of A j. Let h j : A j A j be the j -th power of a right Dehn twist which is affine in the natural Eucliean coorinates on A j. From the efinition of j it

16 16 LAURA DEMARCO AND KEVIN PILGRIM follows that h j extens uniquely to a quasiconformal eformation h j : X(f) X(f) which commutes with f. It follows that the stabilizer of [f] contains the lattice in R N generate by (0,..., 0, j /m j, 0,..., 0), where m j is the moulus of A j, j = 1,..., N. Since in aition the stabilizer of [f] is iscrete, it is a lattice in R N Proof of Theorem 1.2. We now prove that for generic critical heights in the shift locus, the fibers of G are tori coinciing with twist orbits. In other wors, the quotient space T is generically the orbit space for the twist eformation. We begin by characterizing these generic critical heights. A critical height value (h 1,..., h 1 ) is generic for G if (1) h i > 0 for all i, an (2) h i n h j for each i j an all n Z. It is clear that conition (1) correspons to the shift locus, while conition (2) correspons to the stratum S 1 where all critical points lie in istinct foliate equivalence classes. In this stratum, the twisting eformation space has maximal imension 1. Lemma 5.2 implies that f lies in a ( 1)-imensional analytic torus of maps obtaine from twisting. On the other han, the existence of 1 inepenent stretches in the funamental subannuli implies that G has maximal rank 1 along a generic fiber: Lemma 5.3. For each [f] S N, we have rank [f] G N. Proof. The heights map G : S SH is smooth. The locus SH N H is a smooth submanifol of imension N. By Lemma 5.2, for each f, inepenent stretching of the funamental subannuli of f provies a real-analytic analytic section of G over a neighborhoo of G([f]) in SH N an so rank [f] G N. In etail, suppose f represents an element of SH N, an let G be the escape rate of f. For each escaping critical point c, let l(c) be its level: the least integer such that G(f l(c) (c)) G(c 1 ). Choose a critical point c j representing each foliate equivalence class, an relabel them as c 1,..., c N so that G(f l(c j) (c j )) < G(f l(c j+1) (c j+1 )) for all j < N. The mouli m j of the funamental subannuli A j satisfy m 1 = 1 2π G(c 1)

17 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS 17 when N = 1; otherwise, m 1 m 2 (5.2) m 3. m N = 1 2π G(c 1 ) G(f l(c 2) (c 2 )) G(f l(c 3) (c 3 )). G(f l(c N ) (c N )) The matrix has eterminant ( 1) N 1 ( 1), so it is invertible. The mouli m j can be freely ajuste uner analytic stretching eformations. Recalling that G(f k (z)) = k G(z), we see that the rank of G at [f] is at least N. As a consequence of Lemma 5.3, the generic fiber of G is a smooth compact submanifol of real imension 1. Because the twist orbit is both open an close in the fiber, it must coincie with a connecte component. We complete the proof of Theorem 1.2 by observing that the conclusion only hols for the generic heights efine above. Inee, for non-generic critical heights, the imension of the twisting eformation orbit is strictly less than Proof of Theorem 1.5. The existence of the cone structures on spaces B, T, T, an H, an their basic properties follow immeiately from Lemma 5.1 an equation (5.1). 6. The projectivization PT Recall from Theorem 1.5 that there is a well-efine projectivization PT of the via stretching. In this section we prove Theorem 1.6 which states that PT space T is compact an contractible. The proof is very similar to the proof in [6] that the projectivize space of trees PT is compact an contractible. Compactness will follow easily from the properness of G. To prove contractibility, the iea is to construct a eformation retract of PT which lifts the corresponing retract of PT The root point of PT the slice T X(h) T Let X(1) T PT.. Recall that the projectivization PT is ientifie with (1) of T. From Lemma 3.3, for any h 0, there is a unique point in corresponing to polynomials with uniform critical heights equal to h. (1) be this point for height h = 1. We will call this the root point of 6.2. Paths through the shift locus. Assume f lies in the shift locus an has maximal escape rate M(f) = 1. We begin by recalling the construction of [7, Lemma 5.2] which efines a path f t from f to a polynomial with all critical heights equal to 1,

18 18 LAURA DEMARCO AND KEVIN PILGRIM in such a way that for every time 0 t 1, the restriction f {G f > t} is conformally conjugate to f t {G ft > t} an f t satisfies G ft (c) t for all critical points c of f t. The iea is to push the lowest critical values of f up along their external rays until they reach height. This pushing eformation is uniquely efine as long as the critical values o not encounter the critical points of G f. When this occurs, multiple external rays lan at a critical value, so there are finitely many choices for continuing the path. Nevertheless, for any choice an every t > 0, the class [f t ] M lies in the connecte subset of M we calle S(f, t) (cf. the proof of Lemma 3.3 above an [7]). The set S(f, t) is thus containe in a single connecte component of a fiber of the critical heights map G. We will use this pushing-up eformation to efine canonical paths in PT. Lemma 6.1. Suppose {f n } is a sequence of polynomials, f is a polynomial, (f n, X(f n )) (f, X(f)) in B, an t n t in [0, ). If [g n ] S(f n, t n ) an [g n ] [g] in M then [g] S(f, t). Proof. We may assume g n g in P. By hypothesis, for each n, the restriction g n {G gn > t n } is conformally conjugate to f n {G fn > t n }, an G gn (c) t n at all critical points c. Since (f n, X(f n )) (f, X(f)) in B, the restrictions f n {G fn > 0} converge to f {G f > 0} in the Gromov-Hausorff sense. See [7] for etails. In particular, the restrictions f n {G fn > t n } must converge to f {G f > t} (geometrically). It follows that the restriction g {G g > t} is conformally conjugate to f {G f > t}. Furthermore, the critical heights of g n converge to those of g, so [g] S(f, t). Lemma 6.2. Fix an element q T, an let Q be the fiber over q in M. Then for all t > 0, the union Q t = S(f, t) [f] Q is connecte an lies in a unique connecte component of a fiber of the critical heights map G. Proof. By construction, the fiber Q is connecte. Also from the efinitions, the critical heights map G is constant on the set Q t. It suffices to show that Q t is connecte. Recall that for each fixe f, the set S(f, t) is connecte. Let A an B be isjoint open sets such that Q t A B. If Q t has nonempty intersection with both A an B, then at least one of the sets Q A = {[f] Q : S(f, t) A} or Q B = {[f] Q : S(f, t) B}

19 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS 19 is not close. Suppose it is Q A. Since Q = Q A Q B is compact an connecte, there is a sequence [f n ] Q A which converges to [f] Q B. By Lemma 6.1, elements of S(f n, t) A must converge to S(f, t) B. This is a contraiction, so Q t must be connecte Proof of Theorem 1.6. Compactness follows immeiately from the properness of G. Inee, PT is homeomorphic to the slice T (1) of T consisting of points with maximal critical height equal to 1. The slice T (1) is the quotient of the preimage G 1 (1, h 2,..., h 1 ) in M where 1 h 2 h 1 0. By properness of G, this locus is compact, so its quotient in T is also compact. For contractibility, we efine a map R : [0, 1] PT PT by setting R (0, q) = q for all q T (1) an R (t, q) = q t for t > 0, where q t is the unique element in T (1) associate to the set Q t of Lemma 6.2. We observe in the proof of [7, Proposition 4.6] that for any polynomial f, an any family of polynomials f t with [f t ] S(f, t), we have (f t, X(f t )) (f, X(f)) in B as t 0 +. Consequently, the elements q t converge to q in the quotient space PT. Further, the convergence is easily seen to be uniform on a compact neighborhoo of q, by the structure of the open sets which efine the topology of B. It follows that R is continuous at t = 0. It remains to show continuity of R for all t > 0. Suppose q n is a sequence in T (1) converging to q T (1), an choose any sequence t n t in (0, 1]. For each n, choose a polynomial f n so that [f n ] M is in the fiber Q n over q n, an choose g n with [g n ] S(f n, t n ). By compactness, we may pass to a subsequence so that (f n, X(f n )) (f, X(f)) in B. By continuity, f must project to q in T (1). By Lemma 6.1, every accumulation point of the sequence {[g n ]} lies in S(f, t). But uner the projection M PT, the class [g n] projects to (q n ) tn, an all of S(f, t) projects to q t. Therefore, R (t n, q n ) R (t, q), so R is continuous. 7. Simplicial structures an stable conjugacy classes In this section, we prove Theorems 1.7 an 1.8. In particular, we efine a locally finite simplicial complex structure on PST an show that the open simplices of top imension correspon to the topological conjugacy classes of structurally stable polynomials in the shift locus. Our escription of the simplicial structure iffers from the proof in [6] of the analogous statement for PST, since we o not have an intrinsic escription of the points of PST.

20 20 LAURA DEMARCO AND KEVIN PILGRIM 7.1. The open simplices. For each N {1,..., 1}, let σ N = {(x 1,..., x N ) R N : x i > 0 i, x x N = 1} be an open simplex of imension N 1. Fix f in the shift locus an with exactly N critical foliate equivalence classes an maximal critical height M(f) = 1. As escribe in 5.2, the funamental subannuli A 1,..., A N of f can be inepenently stretche, to freely ajust the mouli of the A j. We efine a continuous, injective map σ N S sening (x 1,..., x N ) to the unique point in the stretch-orbit of [f] with mo A j = ( 1) x j. 2π The image of σ N contains [f] an lies in the stratum S N (1) consisting of classes of polynomials in the shift locus with exactly N funamental subannuli an maximal critical height = 1. The critical heights of all maps in the image of σ N can be compute irectly from G(f) an (x 1,..., x N ), using equation (5.2). It follows that the composition σ N S (1) T (1) T (1) H (1) is injective. In fact, the image of σ N is an entire connecte component of the stratum PSH N. We shall see that these σ N fit together to efine the simplicial structure on each of the projectivizations PST PST PSH, an the projection maps are simplicial. We begin by escribing the simplicial structure on PSH in etail Simplicial structure on the space of critical heights. Recall that the projectivization PSH is naturally ientifie with the set of vectors {h = (1, h 2,..., h 1 ) R 1 : 0 < h 1 h 2 h 2 1}. In 1.4 of the Introuction, we introuce a partition PSH = PSH 1 PSH 1 base on the maximal number of inepenent heights in egree. Positive numbers x an y are inepenent if x n y for any integer n. The height vectors in PSH 1 form a iscrete set an will be the 0-skeleton of PSH. Now let h = (1, h 2,..., h 1 ) be a vector with exactly two inepenent heights. Choose h j inepenent from 1, an let n j be the unique integer such that There is a unique continuous map 1 < n j h j <. (0, 1) PSH

21 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS 21 Figure 7.1. The simplicial structure on PSH 4, the space of heights in egree 4, epicte as the triangle {0 < y x 1} in R 2. Though not rawn to scale, the lines represent height relations {x = 1/4 n }, {y = 1/4 n }, an {y = x/4 n } for all positive integers n. with image containing h, sening x (0, 1) to a height vector with exactly two inepenent heights an j-th coorinate equal to h j = 1 + ( 1) x n j Thus, the map is a bijection from the open 1-simplex (0, 1) to the component of PSH 2 containing h; it is easy to see that it extens continuously to a simplicial map [0, 1] PSH 1 PSH 2. In this way, we efine the locally-finite simplicial structure inuctively on strata, so that the union PSH 1 PSHN forms the (N 1)-skeleton of PSH. The simplices are shown for egree 3 in Figure 1.1 an for egree 4 in Figure Proof of Theorem 1.7. Fix a polynomial f with [f] S, with maximal escape rate M(f) = 1, an with N critical foliate equivalence classes. Let σ N = {(x 1,..., x N ) R N : x i 0 i, x x N = 1} be a close simplex of imension N 1. We map the interior σ N to the stretching orbit of [f] within S (1), as escribe above in 7.1. We have alreay seen in 7.2 that the maps σ N PSH N exten to continuous, injective maps σ N PSH.

22 22 LAURA DEMARCO AND KEVIN PILGRIM to efine a simplicial structure on PSH. Because the fibers of T T H are finite in the shift locus, by Lemma 3.2, the maps from σ N exten continuously an injectively through the sequence of spaces σ N PST PST PSH. It follows from [6, 2], an specifically the Proposition 2.17 there, that the (N 1)- imensional open simplices σ N PT either coincie or are isjoint, because the height metrics on a given combinatorial tree are parameterize by σ N. Using the fact that the extene maps σ N PSH efine the simplicial structure on the space of heights, we conclue that the close simplices σ N PT provie the maps for a wellefine simplicial structure on PST, compatible with the projection PST PSH. See [18, Lemma I.2.1]. See also [6, 11] where the simplicial structure on PST 3 is escribe in etail. It remains to show that the images of two open simplices in PST either coincie or are isjoint. Fix a critical height vector h in PSH, an let Q S (1) be a connecte component of a fiber of G over h. The continuity of stretching from Lemma 5.1 implies that the image of Q uner any stretch must remain in a connecte component of a fiber of G. This shows immeiately that the image of simplices either coincie or are isjoint in PST Remark: close simplices in S. While not neee for the proof of Theorem 1.7, it is useful to observe that the open simplices σ N S efine in 7.1 exten continuously an injectively to the closure σ N S. One way to see this is to use the Gromov-Hausorff topology on the space of restrictions (f, X(f)) to the basins of infinity; this topology is equivalent to the topology of uniform convergence on M in the shift locus [7]. The stretching eformation ajusts the relative heights of annuli in the metric on X(f), leaving other invariants of the ynamics of f unchange. In particular, the external angles of rays laning at critical points are fixe, an the collection of polynomials with given critical heights an external angles form a iscrete set in the shift locus. It follows immeiately that the extension to σ N S is well efine an continuous Proof of Theorem 1.8. We conclue this article with the proof that the set of globally structurally stable topological conjugacy classes of maps in the shift locus is in bijective corresponence with the set of top-imensional open simplices of PST. Recall that for maps in the shift locus, quasiconformal an topological conjugacy classes coincie an are connecte. Further, the structurally stable polynomials coincie with the top-imensional stratum S 1 [17]. Let f be a polynomial with 1 inepenent heights in the shift locus an let σ f be the open simplex containing the

23 CRITICAL HEIGHTS ON THE MODULI SPACE OF POLYNOMIALS 23 image of f in PST ; thus σ f is an open simplex of maximal imension. The connecteness of topological conjugacy classes an the efinition of the simplicial structure on PST shows that the assignment f σ f escens to a well-efine function on the set of topological conjugacy classes of such maps. This is clearly surjective. It remains to show injectivity. Suppose σ f1 = σ f2. Then there are elements in the topological conjugacy classes of f 1 an f 2 which lie in the same connecte component of a fiber of G. For generic heights, these connecte components coincie with twisteformation orbits by Theorem 1.2. Therefore f 1 an f 2 are quasiconformally, hence topologically, conjugate. References [1] E. Bierstone an P. D. Milman. Semianalytic an subanalytic sets. Inst. Hautes Étues Sci. Publ. Math. 67(1988), [2] P. Blanchar, R. L. Devaney, an L. Keen. The ynamics of complex polynomials an automorphisms of the shift. Invent. Math. 104(1991), [3] B. Branner an J. H. Hubbar. The iteration of cubic polynomials. I. The global topology of parameter space. Acta Math. 160(1988), [4] B. Branner an J. H. Hubbar. The iteration of cubic polynomials. II. Patterns an parapatterns. Acta Math. 169(1992), [5] R. J. Daverman. Decompositions of manifols. AMS Chelsea Publishing, Provience, RI, Reprint of the 1986 original. [6] L. DeMarco an C. McMullen. Trees an the ynamics of polynomials. Ann. Sci. École Norm. Sup. 41(2008), [7] L. DeMarco an K. Pilgrim. Polynomial basins of infinity. Submitte for publication, [8] L. DeMarco an K. Pilgrim. Hausorffization an polynomial twists. Submitte for publication, [9] L. DeMarco an K. Pilgrim. Escape combinatorics for polynomial ynamics. Preprint, [10] L. DeMarco an A. Schiff. Enumerating the basins of infinity for cubic polynomials. Special Issue of J. Difference Eq. Appl., eicate to Robert L. Devaney. 16 (2010) [11] A. Douay an J. H. Hubbar. Itération es polynômes quaratiques complexes. C. R. Aca. Sci. Paris Sér. I Math. 294(1982), [12] R. Dujarin an C. Favre. Distribution of rational maps with a preperioic critical point. Amer. J. Math. 130(2008), [13] N. Emerson. Dynamics of polynomials with isconnecte Julia sets. Discrete Contin. Dyn. Syst. 9(2003), [14] J. Kiwi. Combinatorial continuity in complex polynomial ynamics. Proc. Lonon Math. Soc. (3) 91(2005), [15] Y. Komori an S. Nakane. Laning property of stretching rays for real cubic polynomials. Conform. Geom. Dyn. 8(2004), (electronic). [16] P. Lavaurs. Systemes ynamiques holomorphes: explosion e points perioiques paraboliques. Thesis, Orsay, [17] C. T. McMullen an D. P. Sullivan. Quasiconformal homeomorphisms an ynamics. III. The Teichmüller space of a holomorphic ynamical system. Av. Math. 135(1998),