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In this paper, we shall prove the conjecture of Inou and Kiwi, using quasiconformal surgery technique. In particular, this shows that a primitive postcritically finite hyperbolic map f0 can be tuned by each g ∈C(T (f0)).
Main Theorem. Let f0 be a postcritically finite hyperbolic polynomial that is primitive and fix an internal angle system for f0. Then the IK straightening map χ : C(f0) → C(T ) is bijective and C(f0) is connected. We shall recall the definition of Inou-Kiwi’s straightening map later. Let us just mention now d that if f0(z)= z + a for some a ∈ C then C(T ) is the set of all monic centered polynomials of degree d which have connected Julia sets. The main part of the proof is to show the surjectivity of the map χ. It is fairly easy to construct a quasi-regular map f which has a generalized polynomial-like restriction hybrid equivalent to a given generalized polynomial in C(T ), via qc surgery in a union of annuli domains around the critical Fatou domainse of f0. In order to show that there is a polynomial map with essentially the same dynamics as f, we run Thurston’s algorithm to f. We modify an argument of Rivera- Letelier [12] to show the convergence. In order to control distortion, we use a result of Kahn [13] on removability of certaine Julia sets. After surjectivity is proved,e we deduce connectivity of C(f0) from that of C(T ) which is a theorem of Branner-Hubbard [1] and Lavaurs [14]. In [3], qc surgery was successfully applied to construct intertwining tuning. Our case is more complicated since the surgery involves the non wandering set of the dynamics. The paper is organized as follows. In §2, we recall definition of generalized polynomial-like maps and the IK straightening map. In §3, we construct a specific Yoccoz puzzle for postcrit- ically finite hyperbolic primitive polynomials which is used to construct renormalizations. In §4, we prove a technical distortion lemma. In §5, we prove a convergence theorem for Thurston algorithm. The proof of the main theorem is given in §6 using qc surgery. Acknowledgment. We would like to thank the referee for carefully reading the paper and in particular, for pointing out an error in Section 3 of the first version. This work was supported by NSFC No. 11731003.
2. The IK straightening map We recall some basic notations and the definition of IK straightening maps. The multi-critical nature of the problem makes the definition a bit complicated. Let Polyd denote the set of all monic centered polynomials of degree d, i.e. polynomials of the d d−2 form z 7→ z + ad−2z + ··· + a0, with a0,a1, ··· ,ad−2 ∈ C. For each f ∈ Polyd, let K(f) and J(f) denote the filled Julia set and the Julia set respectively. Let P (f) denote the postcritical set of f: ∞ P (f)= {f n(c)}. ′ C n=1 f (c)=0[,c∈ [ Let
Cd = {f ∈ Polyd : K(f) is connected}.
2.1. External Rays and Equipotential Curves. For f ∈ Polyd, the Green function is defined as 1 + n Gf (z) = lim log |f (z)|, n→∞ dn where log+ = max(log, 0). The Green function is continuous and subharmonic in C, satisfy- ing Gf (f(z)) = dGf (z). It is nonnegative, harmonic and takes positive values exactly in the PRIMITIVETUNINGVIAQUASICONFORMALSURGERY 3 attracting basin of ∞ n Bf (∞) := {z ∈ C | f (z) → ∞ as n → ∞} = C \ K(f).
Assume f ∈Cd. Then there exists a unique conformal map
φf : Bf (∞) →{z | |z| > 1}
d such that φ(z)/z → 1 as z → ∞ and such that φf ◦ f(z) = (φf (z)) on Bf (∞). The external ray of angle t ∈ R/Z is defined as
−1 i2πt Rf (t)= {φf (re ) | 1 −1 l+i2πθ Ef (l)= {φf (e ) | 0 ≤ θ< 1}. −1 i2πt We say the external ray Rf (t) lands at some point z0 if limr→1 φf (re )= z0. It is known that for each t ∈ Q/Z, Rf (t) lands at some point. On the other hand, if z0 is a repelling or parabolic periodic point, then there exists at least one but at most finitely many external rays landing at z0. See for example [16]. The rational lamination of f, denoted by λ(f), is the equivalence relation on Q/Z so that θ1 ∼ θ2 if and only if Rf (θ1) and Rf (θ2) land at the same point. 2.2. Generalized polynomials over a scheme. Now let us consider a postcritically finite hyperbolic polynomial f0. Following Milnor [17], we define the reduced mapping scheme T (= T (f0)) = (|T |,σ,δ) of f0 as follows. Let |T | denote the collection of critical bounded Fatou components of f0. For r(v) each v ∈ |T |, let r(v) be the minimal positive integer such that f0 (v) is again a critical Fatou component. Define σ : |T | → |T | and δ : |T |→{2, 3, ···} by r(v) r(v) σ(v)= f0 (v) and δ(v) = deg(f0 |v) = deg(f0|v). Definition 2.1 (Generalized Polynomials). A generalized polynomial g over T is a map g : |T |× C → |T |× C such that g(v,z) = (σ(v),gv(z)) where gv(z) is a monic centered polynomial of degree δ(v). Denote by P (T ) the set of all generalized polynomials over the scheme T . Given g ∈ P (T ), the filled Julia set K(g) of g is the set of points in |T |× C whose forward orbits are precompact. The boundary ∂K(g) of the filled Julia set is called the Julia set J(g) of g. The filled Julia set K(g) is called fiberwise connected if K(g, v) := K(g) ∩ ({v}× C) is connected for each v ∈ |T |. Let C(T ) be the set of all the generalized polynomials with fiberwise connected filled Julia set over T . We shall need external rays and Green function for g ∈C(T ) which can be defined similarly as in the case of a single polynomial. Indeed, for each v ∈ |T | there exists a unique conformal ϕv,g : C \ K(g, v) → C \ D such that ϕv,g(z)/z → 1 as z → ∞ and these maps satisfy ϕσ(v),g(gv(z)) = δ(v) −1 2πit ϕv,g(z) . For t ∈ R/Z, the external ray Rg(v,t) is defined as ϕv,g({re : r > 1}). The Green function of g is defined as log |ϕv(z)|, if z 6∈ K(g, v); G (v,z)= g 0 otherwise. 4 WEIXIAO SHEN AND YIMIN WANG 2.3. Generalized polynomial-like maps. We shall now recall the definition of generalized polynomial-like map over the mapping scheme T . We say U is a topological multi-disk in |T |× C if there exist topological disks {Uv}v∈|T | in C such that U ∩ ({v}× C)= {v}× Uv. Definition 2.2. An AGPL (almost generalized polynomial-like) map g over the scheme T is a map ′ g : U → U , (v,z) 7→ (σ(v),gv(z)), with the following properties: ′ ′ • U,U are two topological multi-disks in |T |× C and Uv ( Uv for each v ∈ |T |; v v ′ • g : U → Uσ(v) is a proper map of degree δ(v) for each v; ∞ • The set K(g) := g−n(U), called the filled-Julia set of g, is compactly contained in U. n=0 If in addition U ⋐ U ′, thenT we say that g is a GPL (generalized polynomial-like) map. It should be noted that every AGPL map has a GPL restriction with the same filled-Julia set. See [11, Lemma 2.4]. Let g1,g2 be two AGPL maps over T . We say that they are qc conjugate if there is a fiberwise qc map ϕ from a neighborhood of K(g1) onto a neighborhood of K(g2), sending the v fiber of g1 to the v fiber of g2, such that ϕ ◦ g1 = g2 ◦ ϕ near K(g1). We say that they are hybrid equivalent if they are qc conjugate and we can choose ϕ such that ∂ϕ¯ = 0 a.e. on K(g1). The Douady-Hubbard straightening theorem [5] extends in a straightforward way: every AGPL map g is hybrid equivalent to a generalized polynomial G. In the case that the filled Julia set of g is fiberwise connected, G is determined up to affine conjugacy. For monic centered quadratic polynomials, each affine conjugacy class consists of a single polynomial. For more general maps, it is convenient to introduce an (external) marking to uniquely determine G for a given g. Definition 2.3 (Access and external marking). Let f : U → U ′ be an AGPL map over the mapping scheme T with fiberwise connected filled Julia set. A path to K(f) is a continuous map ′ ′ γ : [0, 1] → U such that γ((0, 1]) ⊂ U \K(f) and γ(0) ∈ J(f). We say two paths γ0 and γ1 to K(f) are homotopic if there exists a continuous map γ˜ : [0, 1] × [0, 1] → U such that (1) t 7→ γ˜(s,t) is a path to K(f) for all s ∈ [0, 1]; (2)γ ˜(0,t)= γ0(t) and γ˜(1,t)= γ1(t) for all t ∈ [0, 1]; (3)γ ˜(s, 0) = γ0(0) for all s ∈ [0, 1]. An access to K(f) is a homotopy class of paths to K(f). An external marking of f is a collection Γ = (Γv)v∈|T | where each Γv is an access to K(f), contained in {v}× C, such that Γ is forward invariant in the following sense. For every v ∈ |T | and every representative γv ⊂ ({v}× C) ∩ U of Γv, the connected component of f(γv) ∩ U which intersects J(f) is a representative of Γσ(v). For a generalized polynomial g ∈ C(T ), there is a standard external marking of g, given by the external rays (R(v, 0))v∈|T | with angle 0. Theorem 1 (Straightening). Let g be an AGPL map over T with fiberwise connected filled Julia set and let Γ be an external marking of g. There is a unique f ∈C(T ) such that there is a hybrid conjugacy between g and f which sends the external marking Γ to the standard marking of f. See [9, Theorem A] for a proof. PRIMITIVETUNINGVIAQUASICONFORMALSURGERY 5 2.4. The Inou-Kiwi map. Let f0,T and r : |T | → N be as in §2.2 and assume f0 is primitive. It is well-known that ∂v is a Jordan curve for each v ∈ |T |. Let C(f0)= {f ∈ Polyd : λ(f) ⊃ λ(f0)}, where λ(f) and λ(f0) are the rational laminations of f and f0 respectively. For each f ∈C(f0), Inou-Kiwi constructed an AGPL (in fact, GPL) map ′ (2.1) F : {v}× Uv → {v}× Uv v v [∈|T | [∈|T | r v over T such that F (v,z) = (σ(v),f ( )(z)) for each z ∈ Uv and such that the filled Julia set of F is the union of {v}× K(v,f): K(v,f)= S(θ,θ′; v) ∩ K(f), θ∼ θ′ λ\(f0) ′ ′ where S(θ,θ ; v) is the component of C\(Rf (θ) ∪ Rf (θ )) which contains external rays Rf (t) for which Rf0 (t) land on ∂v. We shall call such an AGPL map F as in (2.1) a λ(f0)-renormalization ′ of f. While there are many choices of Uv and Uv, the hybrid class of λ(f0)-renormalizations of f is uniquely determined. In order to choose an external marking for F , Inou-Kiwi first fixed an internal angle system which is, by definition, a collection of homeomorphisms α = (αv : ∂v → R/Z)v∈|T | such that: r(v) δ(v)αv = ασ(v) ◦ f0 mod 1. −1 An internal angle system is uniquely determined by the points zv = αv (0), v ∈ |T |, which are (pre-)periodic points of f0. Choose for each v ∈ |T | an external angle θv so that Rf0 (θv) lands at zv. They observed that the external rays Rf (θv) define an external marking of F and this external marking is independent of the choices of θv. Indeed, we can choose (θv)v∈|T | such that δvθv = θσ(v) mod 1 for each v ∈ |T |, see Lemma 2.1. The IK straightening map χ : C(f0) → C(T ) is defined as follows. For each f ∈ C(f0), χ(f) is the generalized polynomial in C(T ) for which there is a hybrid conjugacy from a λ(f0)- renormalization of f to χ(f) sending the external marking determined by the internal angle system to the standard marking for χ(f). Lemma 2.1. Let f0 be as above and let α be an internal angle system. Then there exists r(v) v v −1 v θ ∈ Q/Z, v ∈ |T |, such that Rf0 (θ ) lands at αv (0) and such that f0 (Rf0 (θ )) = Rf0 (θσ(v)). Proof. Claim. Let W be a fixed Fatou component of f ∈Cd and let p be a repelling fixed point of f in ∂U. If the external ray Rf (t) lands at p, then dt = t mod 1. Let Θ ⊂ R/Z denote the set of external angles θ for which Rf (θ) lands at p. It is well known that md : t 7→ dt mod 1 maps Θ onto itself and preserves the cyclic order. So all the angles θ ∈ Θ has the same period under md. We may certainly assume that Θ contains at least two points. Let Γ denote the union of these external rays and {p}. Let V denote the component of −1 C \ Γ which contains W and let V1 denote the component of C \ f (Γ) which contains W . Then ∂V ⊂ ∂V1 and f : V1 → V is a proper map. It follows that f must fix both of the external rays in ∂V . This proves the claim. r(v) −1 −1 −1 Now for each v ∈ |T |, we have f (αv (0)) = ασ(v)(0). So each αv (0) eventually lands at a repelling periodic orbit. The claim enables us to choose θ(v) ∈ R/Z such that δvθv = θσ(v) mod 1 for each v ∈ |T |. 6 WEIXIAO SHEN AND YIMIN WANG 3. Yoccoz puzzle Let f0 be a monic centered postcritically finite hyperbolic and primitive polynomial of degree d ≥ 2. In this section, we shall construct a specific Yoccoz puzzle for f0 (Theorem 2). Recall that Polyd denotes the collection of monic centered polynomials of degree d. We say that a finite set Z is f0-admissible if the following hold: • f0(Z) ⊂ Z, • each periodic point in Z is repelling; • for each z ∈ Z, there exist at least two external rays landing at z. Z Let Γ0 = Γ0 denote the union of all the external rays landing on Z, the set Z itself and the Z −n Z equipotential {Gf0 (z)=1}. For each n ≥ 1, define Γn = f0 (Γ0 ). A bounded component of Z C \ Γn is called a Z-puzzle piece of depth n. The aim of this section is to prove the following: Theorem 2. Let f0 be a monic centered polynomial of degree d ≥ 2 which is postcritically finite, d hyperbolic and primitive. Assume that f0(z) 6= z . Then there exists an f0-admissible finite set Z such that for each (finite) critical point c of f0, if Yn(c) denotes the Z-puzzle piece of depth n ∞ which contains c and U(c) denotes the Fatou component containing c, then n=0 Yn(c)= U(c). We say a point a ∈ J(f0) is bi-accessible if it is the common landing pointsT of two distinct external rays. A point in J(f0) is called buried if it is not in the boundary of any bounded Fatou componnet. We shall need the following lemmas to find buried bi-accessible periodic points. d Lemma 3.1. Let f0 ∈ Polyd be a postcritically finite hyperbolic polynomial with f0(z) 6= z , where d ≥ 2. Then any bi-accessible point in the boundary of a bounded Fatou component is eventually periodic. Proof. Arguing by contradiction, assume that there exists a bi-accessible point a0 which is in the boundary of a bounded Fatou component U and such that a0 is not eventually periodic. By Sullivan’s no wandering theorem, U is eventually periodic. So it suffices to consider the case that U is fixed by f0. ′ ′ ′ Let t,t ∈ R/Z, t 6= t , be such that Rf0 (t) and Rf0 (t ) land at a0. For each n ≥ 1, let n n ′ n ′ ′ an := f0 (a0) and let tn = d t, tn = d t . Then an are pairwise distinct, tn,tn 6∈ Q/Z and ′ ′ ′ tn 6= tn for all n ≥ 0. Let Γn = Rf0 (tn) ∪ Rf0 (tn) ∪{an} and let Wn and Wn be the components ′ of C \ Γn so that Wn ⊃ U and Wn ∩ U = ∅. Note that Wn ∩ U = {an} for each n ≥ 0. Since ∂Wn, n ≥ 0, are pairwise disjoint, Wn, n ≥ 0, are pairwise disjoint. So there exists n0 such that for all n ≥ n0, Wn contains no critical point. Claim. If Wn contains no critical point, then f0(Wn)= Wn+1. −1 To see this, let Γn = f0 (Γn+1) which is a finite union of simple curves stretching to infinity on ′ both sides. Let Vn (resp. Vn) denote the component of C \ Γn which contains an in its boundary ′ ′ ′ ′ and such that Vnb⊂ Wn and ∂Wn ⊂ ∂Vn (resp. Vn ⊂ Wn and ∂Wn ⊂ ∂Vn). Then f0(Vn) ′ ′ ′ ′ and f0(Vn) are distinct components of C \ Γn+1. Since Vnb ⊃ U, we have f0(Vn) = Wn+1 and hence f0(Vn)= Wn+1. Since Wn contains no critical point, f0 : Vn → Wn+1 is a conformal map, which implies that ∂Vn consists of one component of Γn. Thus ∂Vn = ∂Wn, hence Wn = Vn, f0(Wn)= Wn+1. n Thus, for all n ≥ n0, f0(Wn)= Wn+1. It follows thatbf0 (Wn0 )= Wn+n0 for all n ≥ 0. So Wn0 is a wandering domain, which contradicts Sullivan’s no wandering theorem. A more elementary argument is as follows: We can choose θ ∈ Q/Z such that Rf0 (θ) ⊂ Wn0 . As Rf0 (θ) is eventually periodic, Wn0 cannot be a wandering domain. d Lemma 3.2. Let f0 ∈ Polyd be a postcritically finite hyperbolic polynomial with f0(z) 6= z , where d ≥ 2. Then f0 has a bi-accessible repelling periodic point. PRIMITIVETUNINGVIAQUASICONFORMALSURGERY 7 Proof. Without loss of generality, we assume that f0 has a fixed bounded Fatou component U, and let Λ= {t ∈ R/Z : Rf0 (t) lands on ∂U}. Since the Julia set of f0 is locally connected, by the Caratheodory Theorem, there is a continuous surjective map π : R/Z → J(f) such that the external ray Rf0 (t) lands at π(t) and hence −1 π ◦ md = f0 ◦ π, where md : R/Z → R/Z, t 7→ dt mod 1. In particular, Λ = π (∂U) is a non-empty compact subset of R/Z which is forward invariant under the map md. Since d f0(z) 6= z , J(f) is not a Jordan curve, so Λ is a proper subset of R/Z. Thus π :Λ → ∂U is not a homeomorphism. Since π :Λ → ∂U is continuous and surjective, this map is not injective. In other words, there exists a bi-accessible point w ∈ ∂U. By Lemma 3.1, w is eventually periodic. Any periodic point in the orbit of w is a bi-accessible repelling periodic point. d Lemma 3.3. If f0 ∈ Polyd is postcritically finite, hyperbolic and primitive and f0(z) 6= z , then f0 has a buried biaccessible repelling periodic point. Proof. In order to show that f0 has a buried bi-accessible point, it is enough to show that for s some s ≥ 1, f0 has a repelling fixed point which is the landing point of an external ray not s s fixed by f0 . Indeed, if a repelling fixed point p of f0 is in the boundary of a bounded Fatou s component V , then by the assumption that f0 is primitive, we have f0 (U) = U, hence by the s Claim in Lemma 2.1, any external ray landing at p is fixed by f0 . Therefore, it is enough to prove the following Statement N for each N ≥ 0. Statement N. Suppose that f0 ∈ Polyd is a postcritically finite, hyperbolic and primitive d map and f0(z) 6= z , where d ≥ 2. If f0 has at most N attracting periodic points, then there s exists s ≥ 1 such that f0 has a repelling fixed point p which is the landing point of an external s ray which is not fixed by f0 . We proceed by induction on N. Statement 0 is a null statement. Let N ≥ 1 and assume that ′ ′ the statement N holds all 0 ≤ N • all the external rays Rf0 (ti) are fixed by f0; • all attracting periodic points of f0 are fixed by f0. k In particular, f0(p0)= p0. (Note that f0 , k ≥ 1, satisfies the assumption of Statement N.) Let j us construct a Yoccoz puzzle using Z = {p0}. Let Y0 , 1 ≤ j ≤ q denote the puzzle pieces of j j j depth zero and for each n ≥ 1, let Yn denote the puzzle piece of depth n which satisfies Yn ⊂ Y0 j j j and p0 ∈ Yn . Since all the external rays Rf0 (tj ) are fixed, f0 : Yn → Yn−1 is a proper map. Let j j dn,j denote the degree of f0 : Yn → Yn−1, let Dn,j = d1,j d2,j ··· dn,j and let dj = limn→∞ dn,j . j Let ∆n,j = {t ∈ R/Z : Rf0 (t)∩Yn 6= ∅}. Note that ∆0,j is a closed interval and ∆n,j is a disjoint n union of Dn,j closed intervals each of which is mapped onto ∆0,j under md diffeomorphically. j j Let us show d1,j ≥ 2 for each j. Indeed otherwise, f0 : Y1 → Y0 is a conformal map, which implies that md : ∆1,j → ∆0,j is a homeomorphism, which is absurd since ∆1,j intersects both endpoints of ∆0,j. Case 1. Suppose that dj = 1 holds for some j. Let s0 be such that ds,j = 1 for all s>s0. s−s0 j Then f0 |Ys is univalent for all s>s0. Since f0 has only finitely many attracting periodic j points and every attracting cycle of f0 contains a critical point, there exists s1 >s0 such that Ys1 s1 j j does not contain any attracting periodic point of f0. Consider the map f0 : Ys1 → Y0 which has degree D := Ds1,j ≥ d1,j ≥ 2. By the thickening technique ([18]), it extends to a polynomial-like s1 ′ map f0 : Uj → Uj of degree D in the sense of Douady and Hubbard, so it has D fixed points j which are contained in Ys1 . By our choice of s1, none of these fixed point is attracting. Since f0 8 WEIXIAO SHEN AND YIMIN WANG s1 ′ is hyperbolic, it follows that all the D fixed points of f0 : Uj → Uj are repelling. The number j s1 of external rays of f0 which intersect Ys1 and are fixed by f0 is exactly D, with two of them s1 j landing at the same point p0. It follows that one of the repelling fixed point p of f0 |Ys1 is not s1 the landing point of f0 -fixed external ray. Case 2. Assume that dj > 1 holds for all j. Take nj sufficently large so that dn,j = dj for j j all n ≥ nj. Then all critical points of f0|Ynj do not escape from Ynj under iteration, hence j j f0 : Ynj → Ynj −1 is a proper map of degree dj with non-escaping critical points. Again by the ′ thickening technique ([18]), it extends to a polynomial-like map f0 : Uj → Uj of degree dj in the sense of Douady and Hubbard. Thus it is topologically conjugate to a polynomial gj ∈ Polydj with connected Julia set. The polynomial gj is hyperbolic, postcritically finite and primitive. ′ Since for any j′ 6= j, f has an attracting fixed point in Y j , the number of attracting periodic 0 nj′ ′ points of f0 : Uj → Uj is less N. Thus the number of attracting periodic points of gj is less than N. d Subcase 2.1 Assume gj (z) 6= z j for some j. Then by the induction hypothesis, gj has a s periodic pointp ˜ of period s which is not the landing point of gj -fixed external rays. Taking the ′ corresponding periodic point p of f0 : Uj → Uj, we are done. dj ′ Subcase 2.2 Assume gj(z)= z for all j. This implies that the filled Julia set of f0 : Uj → Uj is the closure of a Jordan disk Vj which contains p0. Each Vj is a bounded Fatou component of f0. These bounded Fatou components Vj , 1 ≤ j ≤ q, have p0 as a common point in their closures, contradicting the assumption that f0 is primitive. Proof of Theorem 2. Let Critper denote the set of all periodic critical points of f0 and for each c ∈ Critper, let q(c) denote its period. For each admissible set Z and c ∈ Critper, let sZ (c) ∞ Z sZ (c) denote the minimal positive integer such that f0 (c) ∈ n=0 Yn (c) and let dZ (c) = sZ (c) Z ′ limn→∞ deg(f |Yn (c)). Of course q(c) ≥ sZ (c). Note that if Z ⊂ Z then sZ (c) ≤ sZ′ (c) for T ′ all c ∈ Critper, and if sZ (c)= sZ′ (c) then dZ (c) ≥ dZ′ (c). Given admissible sets Z ⊂ Z we say that Z′ is a (proper) refinement of Z if one of the following holds: • there exists c0 ∈ Critper such that sZ′ (c0) >sZ (c0); • sZ′ (c)= sZ (c) for all c ∈ Critper and there exists c0 ∈ Critper such that dZ′ (c0) < dZ (c0). ∞ Clearly, there does not exist an infinite sequence of admissible sets {Zn}n=1 such that for all n, Zn+1 is a refinement of Zn. Let us say an f0-admissible set Z is buried if Z is disjoint from the boundary of any bounded Fatou component. A buried f0-admissible set exists by Lemma 3.3. It suffices to prove that if Z is a buried f0-admissible set for which the property required by the theorem does not hold, then ′ there exists a buried f0-admissible set Z which is a refinement of Z. ∞ Z To this end, assume that there exists c0 ∈ Critper such that n=0 Yn (c0) ) U(c0). Write s s = sZ (c0). When N is large enough, the critical points of the proper map g = f0 |YN+s(c0) never escapes from its domain. Using the thickening technique ([18])T , g extends to a Douady- Hubbard polynomial-like map with connected Julia set. Thus it is hybrid equivalent to a monic centered polynomial P which is necessarily hyperbolic and postcritically finite. Let D ≥ 2 denote the degree of P and let h denote a hybrid conjugacy. As the filled Julia set of P is not D a topological disk, P (z) 6= z . So by Lemma 3.3, P has a repelling periodic pointz ˆ1 which is −1 biaccessible and buried. By [7, Lemma 3.6] (see also [15, Theorem 7.11]), z1 = h (ˆz1) is a buried ′ biaccessible repelling periodic point of f0. Let Z denote the union of Z and the f0-orbit of z1. ∞ ′ ∞ Z Z ′ ′ As n=0 Yn (c0) is a proper subset of n=0 Yn (c0), either sZ (c0) 6= sZ (c0) or dZ (c0) < dZ (c0). This completes the proof. T T We shall need the following result later. PRIMITIVETUNINGVIAQUASICONFORMALSURGERY 9 Proposition 3.1. Let f0 and Z be as in Theorem 2. Then sup{diam(Y ): Y is a puzzle piece of depth n, Y ∩ Z 6= ∅} → 0 as n → ∞. ∗ Proof. For each n ≥ 0 and z ∈ Z, let Yn (z) denote the union of the closures of the puzzle pieces of depth n which contain z in their boundaries. Since Z is finite, there exists N such that ∗ n ∗ ∗ n YN (z) ∩ P (f0) = ∅ for all z ∈ Z. For each n ≥ 0, and z ∈ Z, f0 : Yn+N (z) → YN (f0 (z)) is a n n ∗ conformal map which extends to a definite neighborhood of f0 (z). It follows that f0 |Yn+N (z) has ∗ uniformly bounded distortion. Since z ∈ J(f), this implies that diam(Yn (z)) → 0 as n → ∞. We shall construct λ(f0)-renormalizations using the puzzle given by Theorem 2. The following is a criterion which will be used in the proof of surjectivity part of the main theorem. Proposition 3.2. Let N0 be a positive integer such that for each v ∈ |T |, the puzzle pieces j r(v) f0 (YN0 (v)), v ∈ |T |, 1 ≤ j ≤ r(v), are pairwise disjoint, and f0 : YN0 (v) → YN0−r(v)(σ(v)) has degree δ(v). Assume that f ∈ Polyd satisfies the following: (1) there is a homeomorphism ψ : C → C with the following properties: • φf ◦ ψ = φf0 holds near ∞, where φf and φf0 are the B¨ottcher map for f and f0 respectively; • ψ ◦ f0(z)= f ◦ ψ(z) for all z ∈ C \ v YN0 (v). (2) The map S F : {v}× ψ(YN0 (v)) → {v}× ψ(YN0−r(v)(σ(v))), v v [ [ defined as F (v,z) = (σ(v),f r(v)(z)), is an AGPL map with fibrewise connected filled Julia set. Then f ∈C(f0) and F is a λ(f0)-renormalization of f. Proof. First of all, note the assumption implies that the filled Julia set K(f) of f is connected and Z = ψ(Z) is an admissible set for f. It suffices to show that f ∈C(f0). Once this is proved, the other statement follows from [9, Proposition 3.13]. Let L(v,f) denote the filled Julia set of F inb the fiber {v}× C. Step 1. We show by induction that for each k ≥ N0, there is a homeomorphsim ψk : C → C −1 which coincide with φf ◦ φf0 on Γk \ J(f). For k = N0, we choose ψN0 = ψ. Assume now that ψk has been defined for some k ≥ N0 and ′ let us construct ψk+1. For each Y ⊂ C, denote Y = ψk(Y ). It suffices to construct, for each ′ Y ∈ Yk, a homeomorphism ψk+1 : Y → Y so that f ◦ ψk+1(z) = ψk ◦ f0(z) for z ∈ Y ∩ Γk+1. Indeed, if Y does not contain a critical point of f0, then f0 : Y → f0(Y ) is a conformalmap, and so ′ ′ ′ −1 is f : Y → f(Y ). In this case, we define ψk+1|Y = (f|Y ) ◦(ψk|f0(Y ))◦(f0|Y )). Assume that ′ Y contains a critical point of f0, so that Y = Yk(v) for some v ∈ |T | and hence Y ⊃ L(v,f). Let ′ ′ B = Yk−r(v)+1(σ(v)), A = Yk+1(v) and X = Yk−r(v)(σ(v)). Then B ⊃ L(σ(v),f), A ⊃ L(v,f) v ′ r( ) r(v) ′ ′ ′ ′ and X ⊃ L(σ(v),f). Since f0 : Y \A → X \B and f : Y \A → X \B are both δ(v)to1 v ′ ′ r( ) r(v) covering, there is a homeomorphism ψk+1 : Y \A → Y \A such that ψk+1 ◦f0 = f ◦ψk on Y \ A and ψk+1 = ψk on ∂Y . Extending the map ψk+1 in an arbitrary way to a homeomorphism ′ ′ from Y to Y , we obtain the desired map ψk+1 : Y → Y . −1 Step 2. For each k ≥ N0, there is a qc map Ψk such that Ψk = φf ◦ φf0 near infinity and such that f ◦ Ψk(z)=Ψk ◦ f0(z) for all z 6∈ v Yk(v). This is well-known. See for example [10, Section 5]. This implies that if Rf0 (θ1) and Rf0 (θ2) land at a common point which is not in ∞ −n S n=0 f0 v∈|T | ∂v (θ1,θ2 ∈ Q/Z), then Rf (θ1) and Rf (θ2) have a common landing point S S 10 WEIXIAO SHEN AND YIMIN WANG as well. Indeed, there exists k such that the whole f0-orbit of the rays Rf0 (θ1) and Rf0 (θ2) lie outside v Yk(v), so Rf (θi)=Ψk(Rf0 (θi)), i =1, 2. Step 3. It remains to show that if Rf0 (θ1) and Rf0 (θ2), θ1,θ2 ∈ Q/Z landing at a common S ∞ −n point in n=0 f0 v∈|T | ∂v , then Rf (θ1) and Rf (θ2) have a common landing point. Let us first assume that the common landing point is in ∂v for some v ∈ |T |. LetΨ=Ψ S S 0 0 N0 be given by Step 2. We define a new qc map H from C\ v v → C\ v L(v,f) such that H =Ψ v outside Y (v) and such that H ◦ f r( ) = f r(v) ◦ H inside Y (v). Note that H maps v N0 0 S v SN0 Rf (θi) onto Rf (θi), i =1, 2. For each v 6= v0, choose a quasidisk Ωv so that these quasidisks are 0 S S pairwise disjoint and disjoint from v ∪ R (θ ) ∪ R (θ ). Let H = H on C \ (v ∪ Ωv), 0 f0 1 f0 2 0 0 v6=v0 and then extend H quasiconformally to C \ v by Beurling-Ahlfors extension. So we obtain 0 0 S a qc map H0 : C \ v0 → C \ L(v0,f) which again maps Rf0 (θi) onto Rf (θi), i = 1, 2. Let −1 ϑ : C \ D → C \ L(v0,f) denote a Riemann mapping. Since ∂v0 is a Jordan curve, ϑ ◦ H0 −1 −1 extends continuously to C \ v0. Since Rf (θi) both land and ϑ (Rf (θ1)) and ϑ (Rf (θ2)) have a common landing point, by Lindelof’s theorem, we conclude that Rf (θi), i = 1, 2, land at the same point. n For the general case, let n ≥ 1 be minimal such that the common landing point of Rf0 (d θi) n (i = 1, 2) lie in v ∂v. As proved above, the external rays Rf (d θi), i = 1, 2, have a common j landing point z. Let k be large integer such that the external rays Rf (d θi), 0 ≤ j 4. Kahn’s quasiconformal distortion bounds In this section, we will modify the argument in [13]1 to obtain a K-qc extension principle. The main result is Theorem 3 which will be used later to show the convergence of the Thurston Algorithm in the proof of the Main Theorem. Throughout we fix a monic centered, hyperbolic, postcritically finite and primitive polynomial d f0 of degree d such that f0(z) 6= z . Let Z be an admissible set given by Theorem 2 and let Z ′ Yn(z) = Yn (z). Let Crit(f0) = {c ∈ C : f0(c)=0} and let Ln denote the domain of the first landing map to Y (c): c∈Crit(f0) n S (4.1) L = z ∈ C : ∃k ≥ 0 such that f k(z) ∈ Y (c) . n 0 n c∈Crit([ f0) Theorem 3. There exists N > 0 and for any puzzle piece Y of depth m ≥0, there is a constant C = C(Y ) > 1 satisfying the following property: if Q : Y → Q(Y ) is a qc map which is conformal a.e. in Y \ Lm+N , then there exists a C-qc map Q : Y → Q(Y ) such that Q = Q on ∂Y . 4.1. Quasiconformal Distortion Bounds and a toy model. The difficulty in proving the e e theorem is that the landing domains Lm+N may come arbitrarily close to the boundary of Y . To deal with the situation, we shall need a toy model developed by Kahn ([13]). Let us first recall some terminology from [13]. Let U ⊂ C be a Jordan domain and A be a measurable subset of U. We say that (A, U) has bounded qc distortion if there exists a constant K ≥ 1 with the following property: if Q : U → Q(U) is a quasiconformal map and ∂Q¯ = 0 a.e. outside A, then there is a K-q.c map Q˜ : U → Q(U) such that Q˜ = Q on ∂U. Let QD(A, U) 1According to Kahn, Yoccoz may have a similar result. PRIMITIVETUNINGVIAQUASICONFORMALSURGERY 11 denote the smallest K satisfying the property. Using this terminology, we can restate Theorem 3 as follows: Theorem 3’. There exists N > 0 such that if Y is a puzzle piece Y of depth m ≥ 0, QD(Lm+N ∩ Y, Y ) < ∞. We shall need the following easy facts. Lemma 4.1. [13, Fact 1.3.6] If A ⊂ U is compact, then QD(A, U) < ∞. Lemma 4.2. [13, Fact 1.3.4] Let U and V be Jordan domains in C and A be a measurable subset of U. If there exists a L-qc map g : U → V and QD(A, U) < ∞, then QD(g(A), V ) ≤ L2QD(A, U). Lemma 4.3. The following statements are equivalent: (i) QD(A, U)= C < ∞; (ii) For any qc map Q : U → Q(U), if Dil(Q) ≤ K for some K ≥ 1 a.e. outside A, then there is a KC-qc map Q˜ : U → Q(U) such that Q˜ = Q on ∂U. Proof. It is obvious that (ii) implies (i). Let us show that (i) implies (ii). Let µ be the Beltrami differential such that µ = ∂Q¯ −1/∂Q−1 on Q(U\A) and µ = 0 otherwise. By the Measurable Rie- mann Mapping Theorem, there exists g : C → C quasiconformal map with Beltrami differential µ. Then g ◦ Q : U → g ◦ Q(U) is a quasiconformal map and ∂g¯ ◦ Q = 0 a.e. outside A. Thus there exists a C-qc map G : U → g ◦ Q(U) such that G = g ◦ h on ∂U, where C = QD(A, U) < ∞. Finally, let Q˜ = g−1 ◦ G and we are done. We shall now recall the recursively notched square model developed in [13]. Let S = (0, 1) × (−1/2, 1/2). Let I denote the collection of the components of (0, 1) \C, where C is the ternary Cantor set. Let (4.2) N = I × [−|I|/2, |I|/2] I [∈I which is a countable disjoint union of closed squares. The following is [13, Lemma 2.1.1]: Theorem 4. QD(N ,S) < ∞. 4.2. Reduce to the toy model. We will work on the polynomial map f0 fixed at the beginning of this section. In the following we write R(θ) for Rf0 (θ). A geometric ray-pair is, by definition, a simple curve consisting of two distinct external rays together with their common landing point. ′ A slice is an open set U bounded by two disjoint ray-pairs R(θi) ∪ R(θi) ∪{ai}, i = 1, 2 such that no external ray lying inside U lands at either a1 or a2. For every z0 ∈ Z, the external rays landing at z0 cut the complex plane C into finitely many sectors S1(z0), ··· ,Sn(z0)(z0). Let S = {Sj(z) | z ∈ Z, 1 ≤ j ≤ n(z)}. We list the elements in S as S1,S2, ··· ,Sν where ν = #S. For each j, the boundary of Sj is a geometric ray-pair: there j − + exists α ∈ Z and θj ,θj ∈ R/Z such that − j + ∂Sj = R(θj ) ∪{α } ∪ R(θj ). − + We order θj ,θj in such a way that − + {t ∈ R/Z : R(t) ⊂ Sj} = (θj ,θj ). Proposition 4.1. For each j ∈{1, 2,...,ν} and each n sufficiently large, there exists a geometric j − + j ray-pair γn = R(tn (j)) ∪ R(tn (j)) ∪{αn} contained in Sj with the following properties: j −n • αn ∈ f0 (Z); 12 WEIXIAO SHEN AND YIMIN WANG − − + + • θj ,tn (j),tn (j),θj lie in R/Z in the anticlockwise order; j • ∂Sj and γn bound a slice; − − + + • tn (j) → θj , tn (j) → θj as n → ∞. We postpone the proof of this proposition to the end of this section and show now how it implies Theorem 3. j j Proof of Theorem 3’. For each n large, let Sn be the slice bounded by γn and ∂Sj given by j j n Proposition 4.1, and let Sn = {z ∈ Sn : G(z) < 1/d }, where G is the Green function of f0. So j b Sn is a finite union of closures of puzzle pieces of depth n. Choose N∗ sufficiently large so that j j b the closure of S := SN∗ is disjoint from the post-critical set of f0. Let q ≫ N∗ be a positive ′ −q j integer so that for any j, j , the diameter of each component of f0 (S ) is much smaller than j′ q j that of S . For each j ∈ {1, 2,...,ν}, f0 maps a neighborhood Qj of α conformally onto the ν j′ q j component of the interior of j′=1 S which contains f0 (αj ). Let Aj = Qj ∩ S . Then Aj is a quasi-disk which contains αj in its boundary and there is σ(j) ∈ {1, 2,...,ν} such that q σ(j) S j j f0 (Aj ) = S . Similarly, there is a quasi-disk Bj which is contained in S and contains αN∗ q τ(j) in its boundary and τ(j) ∈{1, 2,...,ν} such that f0 (Bj )= S . Choosing q large enough, we q + + q − − q − + can ensure that Aj ∩ Bj = ∅. Note that md(θj )= θσ(j),md(θj )= θσ(j), md(tN∗ (j)) = θτ(j) and q + − md(tN∗ (j)) = θτ(j), where md : R/Z → R/Z denotes the map t 7→ dt mod 1. Let ν ν j F : Aj ∪ Bj → S j=1 j=1 [ [ q be the restriction of f0 . Let A = (0, 1/3) × (−1/6, 1/6), B = (2/3, 1) × (−1/6, 1/6) and S = (0, 1) × (−1/2, 1/2) and define a map ν ν G : {j}× (A ∪ B) → {j}× S, j=1 j=1 [ [ as follows: (σ(j), 3z), if z ∈ A; G(j, z)= (τ(j), 3(1 − z)), if z ∈ B. j −q−N∗ Let Cj = {z ∈ S \ (Aj ∪ Bj ): G(z) ≤ d } and C = [1/3, 2/3] × [−1/6, 1/6]. ν j ν Claim. There is a qc homeomorphism H: j=1 S → j=1{j}× S such that (i) H(Aj )= {j}× A, H(Bj )= {j}× B,H(Cj )= {j}× C, ν S S (ii) H ◦ F = G ◦ H holds on j=1 Aj ∪ Bj . ν j ν Indeed, it suffices to prove there isS qc map H0 : j=1 S → j=1{j}× S such that (i) holds and (ii) holds on (∂A ∪ ∂B ) (with H replaced by H ). Indeed, once such a H is constructed, we j j j S 0 S 0 can construct inductively a sequence {H }∞ of qc maps by pull-back which has the following S n n=0 properties: ν j • Hn+1 = Hn on j=1 S \ (Aj ∪ Bj ); • H ◦ F = G ◦ H holds on ν A ∪ B . n Sn+1 j=1 j j These maps Hn have the same maximal dilatation as H0, and they eventually stablize for any ν j S n point in the set X = {z ∈ j=1 S : F (z) 6∈ Aj ∪ Bj for some n}. Since F is uniformly expanding, the set X is dense in Sj, it follows that H converges to a qc map H which S j S n satisfies the requirements. For the existence of H0, a concrete construction of a homeomorphism with the desired properties can be easilyS done using B¨ottcher coordinate with an extra property j j that it is qc in S \Aj ∪ Bj ∪ Cj . It can be made global qc since S , Aj ,Bj , Cj are all quasi-disks. PRIMITIVETUNINGVIAQUASICONFORMALSURGERY 13 Now, let j n Nj = {z ∈ S : ∃n ≥ 1 such that F (z) is well-defined and belongs to Cj′ }. j′ [ Note that for each j, H(Nj )= {j} × N , where N is as in (4.2). Therefore, ν j Q := max QD(Nj ,S ) < ∞. j=1 Let N = q + N . Then any landing domain of Y := Y (c) does not intersect ∗ N c∈Crit(f0) N q−1 ν k j k=0 j=1 f0 (∂Aj ∪ ∂Bj ). Therefore, LN ∩ S ⊂ Nj , so thatS j j j S S QD(LN ∩ S ,S ) ≤ QD(Nj ,S ) ≤ Q. Now let Y be an arbitrary Yoccoz puzzle piece of depth m ≥ 0. Similarly as in the construction for Aj and Bj above, for each x ∈ ∂Y ∩J(f), there is a quasi-disk Vx which is contained in Y and m j(x) contains x in its closure such that f0 maps Vx onto S for some j(x) ∈ {1, 2,...,ν}. Since j(x) S is a finite union of the closure of puzzle pieces of depth N∗ and disjoint from the postcritical k set of f0, for each k = 0, 1,...,m − 1, f (Vx) is a finite union of the closure of puzzle pieces of k depth N∗ + m − k ( j These Vx’s are pairwise disjoint since each of them is mapped onto of a component of j S univalently under f m Noting that (Y ∩ L ) \ V is compactly contained in Y , we 0 m+N x∈∂Y ∩J(f) x S conclude that QD(L ∩ Y, Y ) < ∞. m+N S j j Proof of Proposition 4.1. We denote by Yn the unique puzzle piece of depth n which attaches α j and is contained in the sector Sj . By Proposition 3.1, when n is sufficiently large, Yn is disjoint j j from the postcritical set of f0. Fix n0 large so that for each j, there exists αn0 ∈ Y n0 ∩ J(f0) j − + j with αn0 6∈ Z. Let R(sj ) and R(sj ) be the external rays landing at αn0 which are the boundary j − − + + curves of Yn0 . We assume that θj ,sj ,sj ,θj lie in R/Z in the anticlockwise cyclic order. − + For n ≥ n0, we define two angles tn (j) and tn (j) as following: − − − j tn (j) := sup{θ ∈ (θj ,sj ) | R(θ) ∩ Yn 6= ∅} and + + + j tn (j) := inf{θ ∈ (sj ,θj ) | R(θ) ∩ Yn 6= ∅}. − + Claim 1. For every 1 ≤ j ≤ ν and any n ≥ n0, the two external rays R(tn (j)) and R(tn (j)) j land at the same point, denoted by αn. n −n n − n + Let R (t) = R(t) ∩ {z | Gf0 (z) < d }. Clearly, R (tn (j)) and R (tn (j)) are on the j j − + boundary of Yn and are closest rays (on the boundary of Yn ) to R(sj ) and R(sj ) respectively. − + First, we show that R(tn0+1(j)) and R(tn0+1(j)) land at the same point. j j ± ± Case 1. αn0 ∈ Yn0+1. Then tn0+1(j)= sj , and so claim holds. j j − + Case 2. αn0 6∈ Yn0+1. Then there exists t and t such that ± j j j • R(t ) intersects the boundary of Yn0+1 with a common landing point α 6∈ {αn0 , α }; + − j j • R(t ) ∪ R(t ) ∪{α} separates αn0+1 from α . 14 WEIXIAO SHEN AND YIMIN WANG − − − + + + − Without loss of generality, assume t ∈ (θj ,sj ). Then t ∈ (sj ,θj ). So for any t ∈ (t−,sj ), R(t) is disjoint from Y j , hence t− = tj . Similarly, t+ = tj . Thus t− ∼ t+ . n0+1 n0+1 n0+1 n0+1 λf0 n0+1 The general case can be proved similarly by induction. j n n j It is clear that ∂Sj and γn = R(t−(j)) ∪ R(t+(j)) ∪{αn} bound a slice for each n ≥ n0. So it remains to show − − + Claim 2. The sequence {tn (j)}n>n0 decreases monotonically to θj and {tn (j)}n>n0 increases + monotonically to θj for all j. j j j − Since diam(Yn ) converges to 0, αn → α . Obviously, {tn (j)} is monotonically decreasing, − − − j thus it converges to some θ ∈ [θj ,sj ]. If θ 6= θj , then αn converges to the landing point of R(θ) which is not equal to αj . This leads to a contradiction. 5. Thurston’s Algorithm 5.1. Thurston’s Algorithm. The Thurston algorithm was introduced by Thurston to construct rational maps that is combinatorially equivalent to a given branched covering of the 2-sphere. See [6]. The algorithm goes as follows. Let f : C → C be a quasi-regular map of degree d> 1. Given ∗ a qc map h0 : C → C, let σ0 be the standard complex structure on C and σ = (h0 ◦ f˜) σ0. By e b b ∗ the Measurable Riemann Mapping Theorem, there exists a qc map h1 : C → C with h1σ0 = σ so b −b1 b that Q0 := h0 ◦ f ◦ h1 is a rational map of degree d. The qc map h1 is unique up to composition with a M¨obius transofrmation. Applying the same argument to h1 insteadb ofbh0, we obtain a qc map h2 and a rationale map Q1 of degree d such that Q1 ◦ h2 = h1 ◦ f. Repeating the argument, ∞ ∞ we obtain a sequence of normalized qc maps {hn}n=0 and a sequence of rational maps {Qn}n=0 ∞ of degree d such that Qn ◦ hn+1 = hn ◦ f. The question is to study thee convergence of {hn}n=1 ∞ and {Qn}n=1 after suitable normalization. In [12], Rivera-Letelier applied the algorithme to a certain class of quasi-regular maps which may have non-recurrent branched points with infinite orbits. In this section, we shall modify his argument and prove a convergence theorem for a quasi-regular map f : C → C where the irregular part has a nice Markov structure. For simplicity, we shall assume that f satisfies the following: f −1(∞)=e∞ band fbis holomor- phic in a neighborhood of ∞. Below, we shall use the terminology quasi-regular polynomial for such a map. e e e An open set B is called nice if each component of B is a Jordan disk and f k(∂B) ∩B = ∅ for each k ≥ 1. Let D(B)= {z ∈ C : ∃n ≥ 1 such that f n(z) ∈ B} e denote the domain of the first entry map to B. We say that a nice open set B is free if e P (f) ∩ B = ∅, where e P (f)= {f n(c)}, e n c∈Crit([ f) [≥1 e e and Crit(f) denotes that set of the ramification points of f. We say that an open set B is M-nice if it is nice and for each component B of B, the following three conditions hold: e e (5.1) diam(B)2 ≤ Marea(B); area(B \ D(B)) (5.2) >M −1; area(B) PRIMITIVETUNINGVIAQUASICONFORMALSURGERY 15 (5.3) QD(D(B) ∩ B,B) ≤ M, where QD is as defined in §4. Theorem 5. Let f : C → C be a quasi-regular polynomial of degree d ≥ 2 and let A ⊂ C be a Borel set such that ∂¯f˜ = 0 a.e. outside A. Assume that there is a free open set B which is M-nice for some Me sup w z r |ϕ(w) − ϕ(z)| H(ϕ; z) = lim sup | − |= . rց0 inf|w−z|=r |ϕ(w) − ϕ(z)| So if ϕ is differentiable at z with a positive Jacobian, then |∂ϕ(z)| + |∂ϕ(z)| H(ϕ; z)= . |∂ϕ(z)| − |∂ϕ(z)| Fix K > 1 such that f is K-quasi-regular. Let us call a point z ∈ C regular if the following hold: (1) f n(z) is not a ramificatione point of f for any n ≥ 0; m (2) for any non-negative n and m, the maps hn and f are differentiable at f (z) with a positivee Jakobian. Moreover, e e e H(f; f m(z)) ≤ K. Note that Lebesgue almost every point in C is regular. e e Lemma 5.1. If z is a regular point, then for any integers k>n ≥ 0, the following hold: k−1 ej −1 k−n n j #{n≤j −1 ′ Lemma 5.2. For every k>n, ∂(hk ◦ hn )=0 a.e. on hn(K(n)). Moreover, there exists K -q.c −1 map hk,n such that hk,n = hk ◦ hn on hn(L(n)). e j Proof. For each regular z ∈ K(n), #{n ≤ je < k : f (z) ∈ A} = 0. So by Lemma 5.1, H(hk ◦ −1 hn ; hn(z)) = 1. Since a qc map is absolutely continuous, the first statement follows. Now let us turn to the seconde statement. Note thate for each regular z ∈ L(n), −1 T (5.5) H(hk ◦ hn ; hn(z)) ≤ K , e since by assumption (∗), #{n ≤ j < k : f j (z) ∈ A}≤ T. Put K′ = K4T +1M. ′ W Claim. For each component W of C \ L(n), there is a K -qc map hk,n from hn(W ) onto its W −1e image such that hk,n|∂hn(W )= hk ◦ hn |hn(∂W ). e Once this claim is proved, we can obtain a homeomorphism hk,n : C → C which coincides −1 W with hk ◦ hn outside L(n) and coincides with hk,n on hn(W ) for each component W of C \ L(n). ′ By [5, Lemma 2 in Chapter 1], the map hk,n is K -qc. b b To prove the claim,e let w be the smallest integers such that w ≥ n and f w(W ) ∩ Be 6= ∅. Since B is nice and disjoint from the postcritical set of f, f w maps W homeomorphically onto a component B of B, and f k(W ) ∩B = ∅ for each n ≤ k < w. By the assumptione (∗), for any z ∈ W , #{n ≤ j < w : f j (z) ∈ A}≤ T. By Lemma 5.1, ifez eis regular, then for any n < k ≤ w, e −1 k−n n T e H(hk ◦ hn ; hn(z)) = H(f ; f (z)) ≤ K . −1 T In particular, if n < k ≤ w, then hk ◦ hn is K -qce on hne(W ). Assume now that k > w and let ′ w −1 n −1 w−n T W = (f |W ) (D(B) ∩ B). Since f ◦ hn is conformal on hn(W ) and f is a K -qc map in f n(W ), by Lemma 4.2, we have e e e ′ 2T 2T e QD(hn(W ),hn(W )) ≤ K QD(D(B) ∩ B,B) ≤ K M. For each z ∈ W \ W ′, f w(z) 6∈ D(B), so f j(z) 6∈ B for all j > w. By assumption (∗), it follows that e #{n ≤ j < ke : f j(z) ∈ A}≤ 2T +1. ′ −1 2T +1 Thus for a regular z ∈ W \ W , H(hk ◦ hn ; hn(z)) ≤ K . It follows by Lemma 4.3 that there ′ e −1 is a K -qc map defined on hn(W ) which has the same boundary value as hk ◦ hn . By compactness of normalized K-qc maps and the diagonal argument, there exists a sequence ∞ ′ {kj }j=1 of positive integers such that for each n, hkj ,n converges locally uniformly in C to a K -qc 2 map λn : C → C. Since ∂hkj ,n and ∂λn have bounded norm in L (C) and ∂hkj ,n converges to ∂λn in the sense of distribution, it follows that (5.6) ∂λn = 0 a.e. on hn(K(n)). Lemma 5.3. For each k>n ≥ 0, e λn ◦ hn = λk ◦ hk on L(n). Proof. For each j large enough so that kj >k>n, sinceeL(n) ⊂ L(k), −1 hkj ,n ◦ hn = hkj = hkj ◦ hk ◦ hk =ehkj ,k ◦ ehk is valid on L(n). Letting j go to infinity, we obtain λn ◦ hn = λk ◦ hk on L(n). e e PRIMITIVETUNINGVIAQUASICONFORMALSURGERY 17 5.2. Limit geometry. Let L(n)= λn ◦ hn(L(n)) and K(n)= λn ◦ hn(K(n)). Since we assume A ⊃B, K(n) ⊂ L(n). By Lemma 5.3, for k ≥ n, we have e e L(n)= λk ◦ hk(L(n)) ⊂ λk ◦ hk(L(k)) = L(k) and e e K(n)= λk ◦ hk(K(n)) ⊂ λk ◦ hk(K(k)) = K(k). −1 By (5.6), λn is conformal a.e.on K(n). e e / / / / / ··· CO CO ··· CO CO λn+1 λn λ1 λ0 / Qn / / / Q0 / ··· CO CO ··· CO CO hn+1 hn h1 id f˜ f˜ ··· / C / C / ··· / C / C Lemma 5.4. There exists M ′ > 0 such that for any n ≥ 0 and any component W of C \ L(n), diam(W )2 ≤ M ′area(W ). b −1 w Proof. Let W = (λn ◦hn) (W ). Let w be the minimal integer such that w ≥ n and f (W )∩B= 6 ∅. Then W is a component of C \ L(w) and f w maps W homeomorphically onto a component B of B. Sofϕ := Q0 ◦ Q1 ◦ · · · Qw−1 maps hw(W ) conformally onto B. Moreover,e sincef B has a definitef neighborhood disjointb frome P (f), thee conformalf map ϕ has bounded distortion. Thus 2 ′ diam(hw(W )) /area(hw(W )) is bounded from above.f Since λw are normalized K -qc maps and λw(hw(W )) = W , the statement follows. e f f Lemmaf 5.5. (1) The Lebesgue measure of the set C \ L(n) tends to zero as n → ∞. (2) lim sup{diam(W ): W is a component of C \ L(n)} =0. n→∞ Proof. The second statement follows form the first since all components of C\L(n) have uniformly bounded shape by Lemma 5.4. To prove the first statement, we shall use a martingale type argument.b Let W denote the collection of components of C\L(n), where n runs over all non-negative integers. Let W 0 denote the maximal elements in W , i.e., those that are not contained in any other. For each k ≥ 1, W k W W j W define inductively to beb the maximal elements in \ 0≤j −1 −1 ′ maps λn (W ) conformally onto B and maps λn (W \ ( W ′∈W k+1 W )) onto B \ D(B). Since B has a definite neighborhood disjoint from P (f), ϕ|λ−1(W ) has bounded distortion. Thus n S −1 ′ area(λ (W \ ( ′ W k+1 W )) area(B \ D(B)) C n W ∈ e ≥ C > , −1 area(B) M area(λnS (W )) ′ where C > 0 is independent of W . Since λn are normalized K -qc maps, (5.7) follows. ∞ ∞ Lemma 5.6. n=0 K(n)= n=0 L(n). S S Proof. By the assumption (∗), n K(n)= n L(n). Arguing by contradiction, assume that there exists z ∈ ( L(n)) \ ( K(n)). Then there exists n0 such that for all n ≥ n0,z ∈ L(n) \ K(n). n n S S By definition, there existsz ˜ ∈ L(ne) \ K(n) suche that λ ◦ h (˜z )= z. Then S S n n n n λ ◦ h (˜z )= λ ◦ h (˜z )= z = λ ◦ h (˜z ), n+1 n+1 en en n n n+1 n+1 n+1 soz ˜ =z ˜ . Thereforez ˜ =z ˜ satisfiesz ˜ ∈ ( ∞ L(n)) \ ( ∞ K(n)). This is absurd. n+1 n n0 n=n0 n=0 S S Proof of Theorem 5. By Lemma 5.3, λn ◦ hn = λk ◦ hek on L(n) fore all k>n, we obtain λn ◦ hn(W )= λk ◦ hk(W ) for all component W of C\L(n). By Lemma 5.5, e f fsup |λk ◦ hk(z) − λn ◦ fhn(z)|≤e2 sup diam λn ◦ hn(W ) → 0 z∈C Wf as n tends to ∞, so λn ◦ hn converges uniformlly to a continuous functionf h : C → C. Similarly, −1 −1 sup |λk ◦ Qk ◦ λk+1(z) − λn ◦ Qn ◦ λn+1(z)| z∈C ˜ −1 −1 ˜ −1 −1 = sup |λk ◦ hk ◦ f ◦ hk+1 ◦ λk+1(z) − λn ◦ hn ◦ f ◦ hn+1 ◦ λn+1(z)| z∈C = 2 sup diamW → 0 W −1 as n tends to 0, so λn ◦ Qn ◦ λn+1 converges uniformlly to a proper map f : C → C. −1 ˜ Recall that λn is a normalized K-qc map and conformal Lebsgue a.e. on K(n). By Lemma 5.5 −1 (1) and Lemma 5.6, K(n) = L(n) has full Lebsgue measure. By [12, Lemma B.1], λn n n converges uniformly toS identity, henceS so does λn. We conclude that hn converges uniformly to a continuous map h and Qn converges uniformly to a rational map f of degree d. Thus h ◦ f = f ◦ h. On L(0), h(z) = limn→∞ λn ◦ hn(z) = λ0 ◦ h0(z)= λ0(z). e e 6. Qc surgery and proof of the Main Theorem As before, we fix f0 ∈ Poly(d) which is postcritically finite, hyperbolic and primitive, let T = (|T |,σ,δ) denote the reduced mapping scheme of f0 and let r : |T | → N denote the return r(v) time function. We also fix a collection {θv}v∈|T | of external angles such that d θv = θσ(v) mod 1 and such that Rf0 (θv) lands on the boundary of v, for each v ∈ |T |, according to Lemma 2.1. Choosing two large positive integers N0 ′ ′′ F0 : {v}× Uv → {v}× Uv v v [∈|T | [ ′ r(v) ′ defined by F0|Uv = f0 |Uv, is a GPL map over T , with filled Julia set equal to v{v}× v. We ′ ′′ ′ ′′ may choose these domains Uv and Uv so that Rf0 (θv) intersects ∂Uv (resp. ∂Uv ) at a single v ′ ′ S point. Let U = {z ∈ Uv : F0(v,z) ∈{σ(v)}× Uσ(v)}. Theorem 6. Given g ∈ C(T ) there exists a quasi-regular map f of degree d with the following properties: ′ e (1) f0(z)= f(z) for each z ∈ C \ ( v Uv); ′ (2) There exist quasi-disks Uv,g ⋐ Uv such that f is holomorphic in Uv,g and the map e S ′ r(v) F : {v}× Uv,g → {v}× Uv, (ev,z) 7→ (σ(v), f (z)), v v [ [ is a GPL mape over T which is conformally conjugate to g neare their filled Julia set. More ⋐ ′ ′ precisely, there are quasi-disks Vv,g Vv,g such that g : v{v}× Vv,g → v{v}× Vv,g ′ ′ is a GPL map over T , and for each v ∈ |T | there is a conformal map ϕv : Uv → Vv S S ,g such that ϕv(Uv,g)= Vv,g and r(v) ϕσ(v) ◦ f = g ◦ ϕv holds on Uv,g. (3) Furthermore, if ℓv denote the union of R (θv) \ U ′ and ϕ−1(R (v, 0) ∩ V ′ ), then ℓv e f0 v v g v,g r(v) is a ray, that is a simple curve starting from the infinity, and f (ℓv)= ℓσ(v). ′ ′ Proof. Let av (resp. av) denote the unique intersection point of Rf0 (θv) with ∂Uv (resp. Uv). ′ e Let Vv,g = {z ∈ C : |Gg(v,z)| < 1} and Vv,g = {z ∈ C : |Gg(v,z)| < 1/δ(v)}, where Gg is the ′ Green function of g. Let av,g (resp. av,g) denote the unique intersection point of the external ′ ray Rg(v, 0) with ∂Vv,g (resp. ∂Vv,g). ′ ′ ′ ′ Let ϕv denote the unique Riemann mapping from Uv onto Vv,g such that ϕv(av)= av,g and r v −1 v v v −1 v v ( )−1 ′ −1 v ϕ (a(v)) = a ,g. Define U ,g = ϕv (V ,g) and define f|U ,g = (f |f(Uv)) ◦(ϕσ(v) ◦g◦ϕ ) ′ which is a holomorphic proper map of degree δ(v). Finally, define f on each annulus Uv \ Uv,g e ′ so that f is a quasiregular covering map from this annulus to f0(Uv \ Uv) of degree δ(v) and f −1 ′ ′ maps the arc ϕv (Rg(v, 0) ∩ (Vv \ Vv)) onto the arc f0(Rf0 (θv) ∩e(Uv \ Uv)). All the desired propertiese are easily checked. e Proof of the Main Theorem (Surjectivity). Let f be the quasi-regular map constructed in Theo- rem 6, and let C ≥ 1 be the maximal dilatation of f. Let us check that it satisfies the conditions of Theorem 5. Firstly, it is a quasi-regular polynomiale and ∂f¯ = 0 a.e. on C \ A, where e ′ A := Uv\Uv,g ⊂ YN0+N (v). v v [∈|T | [ To construct the set B, let R(A)= {x ∈ A : ∃n ≥ 1 such that f n(z) ∈ A} e 20 WEIXIAO SHEN AND YIMIN WANG be the return domain to A under f. For every x ∈ R(A), there is a smallest integer k = k(x) ˜k such that f (x) ∈ YN0 (v)\YN0+r(v)(v)=:Ω. It is easy to see Q := sup k(x) < ∞. Let v e x∈R(A) S n E = z ∈ Ω: ∃n ≥ 1, such that f (z) ∈ YN0 (v) , ( v ) [ and let B be the union of components of D(E) which intersecte R(A). So B⊃ R(A). Consequently, if f j (z) 6∈ B for 0 ≤ j Proof. The proofs in the literature were stated for the case C(d). For d = 2 this is due to Douady-Hubbard ([4]), the case d = 3 was proved by Branner-Hubbard ([1]) and for all d ≥ 3 this was proved by Lavaurs ([14]). The proof of Lavaurs generalizes to the case of C(T ) in a straightforward way. See also [2] for a stronger result with a different proof. Proof of the Main Theorem (connectivity). We shall show that if E is a non-empty open and closed subset of C(f0), then χ(E) is a closed subset of C(T ). Together with connectivity of C(T ) and bijectivity of the map χ, this implies that C(f0) is connected. Suppose that gn is a sequence in χ(E) and gn → g in C(T ). We need to show that g ∈ χ(E). −1 Let fn = χ (gn) ∈ E. Since E is compact, passing to a subsequence we may assume that fn → f ∈ E. As in [5, Section 7], we may choose hybrid conjugacies hn between λ(f0)-renormalization of fn and gn so that the maximal dilatation of hn is uniformly bounded. Passing to a further subsequence, we see that the λ(f0)-renormalization of f is qc conjugate to g respecting the external markings. Thus f is conjugate to χ−1(g) via a qc map h : C → C which is conformal outside the filled Julia set of f and satisfies h(z) = z + o(1) near infinity. The Beltrami path connecting f and χ−1(g) is contained in E and thus g = χ(χ−1(g)) ∈ χ(E). References [1] B. Branner, J. Hubbard. The iteration of cubic polynomials Part I: The global topology of parameter space. Acta Math., 160 (1988), 143-206. [2] L. De Marco and K. Pilgrim. Polynomial basins of infinity. Geom. Funct. Anal. 21 (2011), 920-950. [3] A. Epstein and M. Yampolsky. Geography of the cubic connectedness locus: intertwining surgery. In Ann. Sci. Ecole Norm. Sup. 32, 151-185. [4] A. Douady and H. Hubbard, Etude´ dynamique des polynˆomes complexes. Publications Math´ematiques d’Orsay, Universit´ede Paris-Sud, D´epartement de Mathematiques, Orsay, 1984 [5] A. Douady and H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Ecole Norm. Sup. (4) 18 (1985) 287-343. [6] A. Douady and H. Hubbard. A proof of Thurston’s topological characterization of rational functions. Acta Math., 171(2), 263-297. [7] H. Inou, Renormalization and rigidity of polynomials of higher degree, J. Math. Kyoto Univ. 42 (2002) 351- 392. [8] H. Inou, Combinatorics and topology of straightening maps II: discontinuity, arXiv:0903.4289v2. [9] H. Inou, J. Kiwi, Combinatorics and topology of straightening maps I: compactness and bijectivity. Adv. Math. 231 (2012), 2666-2733 [10] O. Kozlovski, W. Shen, S. van Strien. Rigidity for real polynomials, Ann. Math., 165 (2007), 749-841 [11] M. Lyubich and M. Yampolsky. Dynamics of quadratic polynomials: complex bounds for real maps. Ann. Inst. Fourier 47 (1997), 1219-1255. [12] J. Rivera-Letelier, A connecting lemma for rational maps satisfying a no growth condition. Ergodic Theory Dynam. System 27 (2007),no.2, 595-636. [13] J. Kahn, Holomorphic Removability of Julia Sets. arXiv:math/9812164 [14] P. Lavaurs. PhD Thesis at University of Paris Sud, Orsay, 1989. [15] C. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994 [16] J. Milnor, Dynamics in one complex variable. Third Edition, Annals of Mathematics Studies 160, Princeton University Press, 2006. [17] J. Milnor, Hyperbolic components, in: Conformal Dynamics and Hyperbolic Geometry, in: Contemp. Math., vol. 573, Amer. Math. Soc., Providence, RI, 2012, pp. 183-232. [18] J. Milnor, Local connectivity of Julia sets: expository lectures. The Mandelbrot Set, Theme and Variations, volume 274 of London Math. Soc. Lecture Note Ser., 67-116. Cambridge Univ. Press, Cambridge. [19] P.Roesch, Y. Yin.The boundary of bounded polynomial Fatou components. C. R. Math. Acad. Sci. Paris, 346, 877-880. [20] M.Shishikura, On the quasiconformal surgery of rational functions. Ann. Sci. Ecole Norm. Sup. 20 (1987), 1-29. 22 WEIXIAO SHEN AND YIMIN WANG Shanghai Center for Mathematical Sciences, Fudan University, No. 2005 Songhu Road, Shanghai 200438, China Email address: [email protected], [email protected] area(B). This proves that B is M-nice.e e So by Theorem 5, there is a continuous surjective map h and a map f ∈ Polyd such that f ◦ h = h ◦ f. The map h is holomorphic and h(z) = z + o(1) near infinity. Near ∞, f = f0. Thus h = φ ◦ φ−1 near ∞, where φ and φ are the B¨ottcher map for f and f respectively. It f f0 f f0 0 follows that eh(ℓv) is the external ray Rf (θv). By Proposition 3.2, f ∈ C(f0) and F : ve{v}× r(v) h(Y v (v)) → {v}× h(Y (v)), F |h(Y v (v)) = f , is a λ(f )-renormalization of N0+r( ) v N0 N0+r( ) 0 S f. S In order to show that χ(f) = g, we need to show that F and F are hybrid equivalent. Let us consider the associated maps F : v YN0+r(v)(v) → v YN0 (v) and F : v h(YN0+r(v)(v)) → r(v) r(v) h(Y (v)), where F|Y v (v)= f |Y v (v) and F|h(Y e v )= f |h(Y v (v)). v N0 N0+r( ) S N0+r( ) S N0+r( ) S N0+r( ) It suffices to show that there is ae qc map H : U ′ → h(U ′ ) such that H ◦ F = F ◦ H and S v v v v such that H¯ = 0 a.e.e on the filled Juliae set K(F) of F. Note that h ◦ F = F ◦ h and K(F) S S contains the postcritical set of F. By Theorem 5, there is a qc map λ0 such that eλ0 = h outside ′ ∞ −k ¯ e e n e e B := k=0 f (B) and such that ∂λ0 = 0 a.e. on {z : f (z) ∈/ A, ∀n ≥ 0}. In particular, ∂λ¯ = 0 a.e. on K(F). Since Be′ is a countable union of Jordan disks with pairwise disjoint clo- 0 S sure and it ise disjoint from K(F), λ0 is homotopic to h rel Ke(F). Therefore, there is a sequence e of qc maps λk : v YN0 (v) → v h(YN0 (v)), k =1, 2,..., all homotopic to h rel K(F), such that F ◦ λ = λ ◦ F and λ = λ eon W := (Y (v) \ Y v (ev)) holds for k =0, 1,.... Since F k+1 k S k 0S v N0 N0+r( ) is holomorphic, F is holomorphic outside A, and each orbit of F passes through A ate most once, S the maximal dilatatione of λk is uniformly bounded. Since λk(z) eventually stablizes for each z in the domain ofeF, λk converges to a qc map H. Moreover, ∂H¯e = 0 holds a.e. on K(F) since so does λk for each k. e e In order to show that C(f0) is connected, we shall make use of the following result. Theorem 7 (Branner-Hubbard-Lavaurs). The set C(T ) is a connected compact set. PRIMITIVETUNINGVIAQUASICONFORMALSURGERY 21