Distortion control of conjugacies between quadratic polynomials

SCIENCE CHINA 

Mathematics 

. ARTICLES . March 2010 Vol. 53 No. 3: 625–634 

doi: 10.1007/s11425-010-0041-7 

Distortion control of conjugacies 

between quadratic polynomials 

Dedicated to Professor Yang Lo on the Occasion of his 70th Birthday 

CUI GuiZhen 1,∗ &TANLei 2 

1 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; 

2 Laboratoire de Mathématiques, UMR du CNRS 6093, Université d’Angers, 2 bd. Lavoisier, 

49045 Angers cedex 01, France 

Email: gzcui@math.ac.cn, tanlei@math.univ-angers.fr 

Received September 25, 2009; accepted December 15, 2009 

Abstract We use a new type of distortion control of univalent functions to give an alternative proof of 

Douady-Hubbard’s ray-landing theorem for quadratic Misiurewicz polynomials. The univalent maps arise from 

Thurston’s iterated algorithm on perturbation of such polynomials. 

Keywords distortion, conjugacy, polynomial 

MSC(2000): 37F10, 30C35 

Citation: Cui G Z, Tan L. Distortion control of conjugacies between quadratic polynomials. Sci China Math, 2010, 

53(3): 625–634, doi: 10.1007/s11425-010-0041-7 

1 Introduction 

There are many quantities to measure the distance of a univalent function f from Möbius transformations 

besides of the C 0 -topology, for example, the Schwartz derivative Sf , or the nonlinearity f ′′ /f ′ .In[3],we 

introduce a new type of distortion control and prove an a priori bound of the distortion by applying this 

new type of distortion control. Another version of this a priori bound theorem is in particular used for 

complex dynamics. In this work, we use this theorem to give an alternative proof of Douady-Hubbard’s 

ray-landing theorem for quadratic Misiurewicz polynomials. 

1.1 Definition of the distortion 

Let E ⊂ C be an open set and φ : E↩→ C be a univalent holomorphic function. Define 

D(φ, E) =sup{|mod(φ(A)) − mod(A)|}, 

where A ⊂ C are annuli with finite moduli and ∂A ⊂ E, and by abuse of notation φ(A) denotes the 

annulus in C bounded by φ(∂A) (the map φ might not be defined on some points of A). 

∗ Corresponding author 

c○ Science China Press and Springer-Verlag Berlin Heidelberg 2010 math.scichina.com www.springerlink.com

626 CUI GuiZhen et al. Sci China Math March 2010 Vol. 53 No. 3 

1.2 A priori bound 

Let g be a geometrically finite rational map, i.e., the post-critical set Pg (the closure of the union of all 

the critical orbits) has a finite (or empty) accumulation set. Assume that the Fatou set of g is non-empty. 

Let X0 be the union of finitely many periodic cycles contained in the Julia set of g. It is known that every 

point in X0 is either parabolic or repelling. Set X1 = g−1 (X0)\X0, Xn+1 = g−n (X1) andX = 

n0 Xn. 

Now for each y ∈ X0, chooseUyasimply connected neighborhood of y satisfying the following properties: 

(1) These domains Uy are disjoint pairwise. 

(2) For each n 1andeachpointx∈Xnwith gn (x) =y ∈ X0, letUxbe the connected component of 

g−n (Uy) containing x, and then the map gn : Ux \{x} →Uy \{y} is a covering (this implies that Ux ∩ Pg 

is either empty or equal to {x}). 

Denote by D the unit disc. For any point y ∈ X0, there is a Riemann mapping χ from Uy to D with 

χ(y) =0. SetUy(r) =χ−1 ({z : |z| 0 and a positive function C(r) defined on (0,r0) with 

C(r) → 0(as r → 0) such that for any n 0 and any univalent holomorphic map φ : Vn(r) ↩→ C, 

1.3 External rays 

D(φ, E) C(r). 

Now let us consider quadratic polynomials Qc(z) =z 2 + c. DenotebyKc the filled-in Julia set of Qc. For 

any c ∈ C, there exists a conformal map φc defined on a neighborhood of the infinity such that φc(∞) =∞, 

φc(z)/z → 1asz →∞and φc ◦ Qc(z) =(φc(z)) 2 .Themapφc is called the Böttcher coordinate of Qc at 

the infinity. If Kc is connected, then φc defines a Riemann mapping φc : C \ Kc → C \ D. Set 

Rc(θ) ={φ −1 

c (re 2πiθ ),r >1}. 

It is called the dynamical θ-ray for Qc. Even if Kc is disconnected, φc can be extended to a conformal 

map from a domain U whose boundary passes through the critical point zero to {z : |z| >r} for some 

constant r>1. In particular, the critical value c is contained in U. 

Recall that the Mandelbrot set is defined by M = {c : {Qn c (c)}n0 is bounded}. Equivalently, the point 

c is contained in M if and only if Kc is connected. Define Φ(c) =φc(c). It turns out that Φ : C\M → C\D 

is a conformal map with Φ(z)/z → 1asz →∞(refer to [4]). Set RM (θ) ={Φ−1 (re2πiθ ),r > 1}. It is 

called the parameter θ-ray. 

Theorem 1.2 (Pre-periodic external rays landing). Let c ∈ M be a parameter such that a strictly 

pre-periodic dynamical ray Rc(θ) lands at c. Then the parameter ray RM(θ) lands also at c. 

Such a polynomial Qc is called a Misiurewicz polynomial. This theorem is proved by Douady and 

Hubbard using a perturbation argument (refer to [5, p. 94]). We will reprove this theorem in this paper. 

Our approach is not known in the literature. 

2 Various types of distortions 

Let E ⊂ C be an open set and φ : E↩→ C be a univalent holomorphic function with D(φ, E) < ∞. The 

next lemma is easy to verify.

CUI GuiZhen et al. Sci China Math March 2010 Vol. 53 No. 3 627 

Lemma 2.1. (a) D(φ −1 ,φ(E)) = D(φ, E). 

(b) D(γ ◦ φ ◦ β,β −1 (E)) = D(φ, E), for any Möbius transformations β and γ of C. 

(c) D(φ, E1) D(φ, E) if E1 ⊂ E. 

(d) Assume that φn : E↩→ C is a sequence of univalent functions that converges locally uniformly to a 

univalent function φ. ThenD(φ, E) lim infn→∞ D(φn,E). 

2.1 Hyperbolic sup-norm of the Schwarzian derivative 

Let E ⊂ C be a hyperbolic open set. Denote by ρE(z) thePoincarédensityofE. Let φ : E↩→ C be a 

univalent function. The Schwarzian derivative of φ is defined by 

Sφ(z) = φ′′′ (z) 

φ ′ (z) 

− 3 

2 

′′ φ (z) 

φ ′ 2 . 

(z) 

Theorem 2.2. There is a universal constant C1 > 0 such that 

sup |Sφ(z)|ρE(z) 

z∈E 

−2 

C1 D(φ, E). (1) 

To prove this theorem, we need the following lemmas. Refer to [3] for the proof of the next lemma. 

Lemma 2.3. Let E ⊂ C be a hyperbolic open set with 0, 1, ∞∈E. Let φ : E↩→ C be a univalent 

function with 0, 1 and ∞ fixed. Then | log |φ ′ (z)|| 5πD(φ, E) for any point z ∈ E \{∞}. 

Lemma 2.4. There is a universal constant C2 > 0 such that 

sup |Sφ(z)|ρE(z) 

z∈E 

−2 C2. (2) 

This lemma is a generalization of the classical theorem of Kraus-Nehari (see below), and is first proved 

by Beardon and Gehring (see [1]) with the explicit constant C2 = 3. Here we give an independent proof 

using only the estimation of Poincaré densityandKöebe distortion theorem. The following results are 

known: 

(A) (refer to Theorem 2.1 in [6]) For any annulus A ⊂ C with mod(A) > 5log2/(2π), there exists an 

essential round annulus B ⊂ A (i.e., B separates the boundary components of A) such that 

mod(B) mod(A) − 5log2/(2π). (3) 

(B) (Kraus-Nehari theorem, refer to [6, p. 60]) Let D be a round disc in C with Poincaré densityρD 

and f : D↩→ C be a univalent function. Then 

sup |Sf (z)|ρ 

z∈D 

−2 

D (z) 3/2. (4) 

We say that two positive quantities ρ1 and ρ2 are comparable (denoted by ρ1 ∼ ρ2) if there is a universal 

constant C>1 such that 1/C < ρ1/ρ2


Proof. Fix a constant r with 32


For any z ∈ E, choosez1 ∈ E such that ρE1(z) 2ρE(z) forE1 := E \{z1}. ChoosenowtwoMöbius 

transformations β and γ such that β(z1) =∞, β(z) =1,d(1,∂β(E1)) = 2 such that ψ := γ ◦ φ ◦ β −1 

fixes 0, 1and∞. SetG := β(E1). It is contained in C. Wehave 

|Sφ(z)| 

ρ 2 E 

|Sφ(z)| 

4 =4|Sψ(1)| . (8) 

(z) (z) (1) 

ρ 2 E1 

Since δ 0. Thus 

|ψ ′ (z) − ψ ′ (1)| C4δ 

for |z − 1| 1. Now we may apply the Cauchy integral formula to ψ ′ (z) onthecircle|z−1| =1 

to get that |ψ ′′ (1)|, |ψ ′′′ (1)| C5δ. Combining with ψ ′ |Sψ(1)|ρG(1) 

(1) ∈ Aδ, we get |Sψ(1)| C6δ. Therefore 

−2 C7δ(1 + | log δ|) 2 . 

Case B. mod(1,G) > | log δ|/π > 6log2/π. 

Let A be an essential round annulus in G whose core curve passes through the point 1 with modulus 

equal to mod(1,G), which is equal to log R/π for some constant R>0. Then 

So R>1/δ > 64. From Lemma 2.5, we have 

|Sψ(1)| 

ρ 2 G (1) 

Case 2 

ρ 2 G 

log R/π =mod(A) =mod(1,G) > | log δ|/π. 

12 |Sψ(1)| 2 log R 

∼ 

(1) π(R − 32) 

ρ 2 A 

Hence in both cases (9), (1) is proved. 

2.2 Controlling distortions of conjugations 

2 

 

2 

12 2 | log δ| 

2} and hence φc is univalent in E = {z : |z| > 8}∪{∞} by Koebe distortion 

theorem. 

Lemma 2.6. There is a universal constant C0 > 0, such that for any points c1,c2 ∈ C \ M with 

|Φ(ci)| 2(i =1, 2), we have 

 

|c1 − c2| C0 D(ψ, E), 

where ψ = φ−1 ◦ φc1. 

c2 

Proof. Let φc(z) =z+b0+b1/z+··· be the expansion at the infinity. By the formula φc◦Qc(z) =φc(z) 2 , 

one may check that b0 =0andb1 = c/2. Therefore ψ(z) has the expansion ψ(z) =z +(c1−c2)/(2z)+··· at the infinity. 

Let Ψ(z) =1/ψ(1/z) =z + a2z2 + a3z3 + ···. Using ψ(1/z)Ψ(z) ≡ 1 one obtains that a2 =0and 

a3 =(c2− c1)/2. This implies that SΨ(0) = 6a3 =3(c2−c1). Let ˜ρ(z)|dz| be the Poincaré metricon 

{z : |z| < 1/8}. Then˜ρ(0) = 16. Denote by ρ(z)|dz| the Poincaré metriconE. Then 

Sψ(z) 

lim 

z→∞ ρ2 SΨ(0) 

= 

(z) 

˜ρ 2 (0) = 3(c2 − c1) 

. 

256


Therefore by Theorem 2.2, 

3 Thurston algorithm 

|c1 − c2| = 256 

3 

|Sψ(z)| 

lim 

z→∞ ρ2 

C0 D(ψ, E). 

(z) 

We will apply our distortion control to univalent maps arising naturally from the Thurston algorithm of 

perturbations of rational maps. 

3.1 c-Equivalence between semi-rational maps 

Let F : C → C be a branched covering with deg F 2. Its post-critical PF is defined to be the closure of 

the forward orbits of the critical points. Denote by P ′ F the accumulation set of PF . We say that F is a 

sub-hyperbolic semi-rational map if P ′ F is finite and either P ′ F = ∅ or F is holomorphic in a neighborhood 

of P ′ F and each cycle in P ′ F is either attracting or super-attracting. 

Two sub-hyperbolic semi-rational maps F1 and F2 are c-equivalent if there is a pair of homeomorphisms 

(φ0,φ1) ofC such that (a) φ0 ◦ F1 = F2 ◦ φ1; (b) the two maps φ0 and φ1 are isotopic rel PF1; and(c) 

φ0 is holomorphic in a neighborhood of P ′ F1 (hence φ1 is also holomorphic and coincides with φ0 in a 

neighborhood of P ′ F1 ). 

3.2 Thurston algorithm 

Let F : C → C be a sub-hyperbolic semi-rational map with P ′ ′′ 

F non-empty. Denote by P F the union of 

P ′ F with all the periodic cycles in PF which meet critical points. Assume that F is holomorphic in a 

neighborhood of P ′′ 

F . Pick three distinct points in PF . In this section, we say that a homeomorphism of 

C is normalized if it fixes these three points. 

Assume that the sub-hyperbolic semi-rational map F is c-equivalent to a rational map f via a pair 

of normalized homeomorphisms (φ0,φ1). Since F is holomorphic in a neighborhood of P ′′ 

F ,thereisa 

pair of normalized homeomorphisms (ξ0,ξ1) ofC in the isotopy class of φ0 rel PF , such that they are 

holomorphic and coincide with each other in a neighborhood of P ′′ 

F ,andξ0◦F = f ◦ ξ1. Furthermore, 

for n 2, there is a normalized homeomorphism ξn of C in the isotopy class of ξ0 rel PF such that 

ξn−1 ◦ F = f ◦ ξn and ξn coincides with ξ0 in a neighborhood of P ′′ 

F . 

Lemma 3.1. The sequence {ξn} is uniformly convergent. 

This lemma is proved by Shishikura and Rees in the case that PF is finite (refer to [7]). One may check 

that their proof works in the case that F is a sub-hyperbolic semi-rational map. 

Assume that η0 is a normalized homeomorphism of C such that η0 is holomorphic in a neighborhood 

of P ′′ 

F . Then there is a unique normalized homeomorphism η1 of C such that f1 := η0 ◦ F ◦ η −1 

1 is 

holomorphic. Obviously, η1 is also holomorphic in a neighborhood of P ′′ 

F . Recursively, there is a unique 

normalized homeomorphism ηn of C such that fn := ηn−1 ◦ F ◦ η−1 n is holomorphic. Then ηn is also 

holomorphic in a neighborhood of P ′′ 

F . The sequence of rational maps {fn} is called a Thurston sequence 

of F . 

Theorem 3.2 (Convergence of the Thurston algorithm). Assumethesub-hyperbolicsemi-rational 

map F is c-equivalent to a rational map f via a pair of normalized homeomorphisms. Then the Thurston 

sequence {fn} converges uniformly to the rational map f. Moreover, the sequence {ηn} is also uniformly 

convergent. 

Proof. By the definition we know that (a) φ0 ◦ F = f ◦ φ1; (b) the two maps φ0 and φ1 are isotopic 

. Since F is holomorphic in a 

rel PF ;and(c)bothφ0 and φ1 are holomorphic in a neighborhood of P ′ F 

neighborhood of P ′′ 

F ,wemaymodifyφ0initshomotopy class such that both φ0 and φ1 are holomorphic 

and coincide in a neighborhood of P ′′ 

F . As above, we have a sequence of normalized homeomorphisms


{φn}n1 of C in the isotopy class of φ0 rel PF , such that φn−1 ◦F = f ◦φn and φn = φ0 in a neighborhood 

of P ′′ 

F . 

Set ψn = ηn ◦φ−1 n .Thenψnis holomorphic in a neighborhood of P ′′ 

f . We may assume furthermore that 

ψ0 is a K-quasiconformal map of C by modifying φ0 in its isotopy class. Then ψn is also K-quasiconformal. 

It is easy to check that 

ψn−1 ◦ f = fn ◦ ψn. 

See the following diagram: 

··· → C 

f 

−→ C → ··· → C 

f 

−→ C 

f 

−→ C 

↑ φn ↑ φn−1 ↑ φ2 ↑ φ1 ↑ φ0 

··· → C 

F 

−→ C → ··· → C 

F 

−→ C 

F 

−→ C 

↓ ηn ↓ ηn−1 ↓ η2 ↓ η1 ↓ η0 

··· → C 

fn 

−→ C → ··· → C 

f2 

−→ C 

f1 

−→ C . 

Since ψ0 is a K-quasiconformal map of C and is holomorphic in a neighborhood W of P ′′ 

f (we may choose 

W such that it is contained in the Fatou set of f and f −1 (W ) ⊃ W ), by the equation ψn−1 ◦ f = fn ◦ ψn, 

we see that ψn is also a K-quasiconformal map of C and holomorphic in f −n (W ). Therefore there is 

a subsequence of {ψn} which converges uniformly to a limit quasiconformal map ψ of C. Moreover, ψ 

is holomorphic in ∪f −n (W ) which is the Fatou set. Thus ψ is holomorphic on C since the measure of 

the Julia set of f is zero. Combining with the normalization condition, we know that ψ is the identity. 

Therefore the entire sequence {ψn} converges uniformly to the identity. It follows that the Thurston 

sequence {fn} converges uniformly to the rational map f . 

Because ηn = ψn ◦ φn, from Lemma 3.1, we know that the sequence {φn}, and therefore the sequence 

{ηn} is uniformly convergent. 

4 Misiurewicz-hyperbolic deformation 

We now begin to prove Theorem 1.2. Recall that φc is the Böttcher coordinate of the quadratic polynomial 

Qc(z) =z 2 + c at infinity. For c ∈ M, themapφc defines a Riemann representation φc : C \ Kc → C \ D. 

The dynamical θ-ray is defined by 

Rc(θ) ={φ −1 

c (re 2πiθ ),r >1}. 

For the Mandelbrot set M, the Douady-Hubbard Riemann representation Φ : C \ M → C \ D is defined 

by Φ(c) =φc(c). The parameter θ-ray is defined by 

RM (θ) ={Φ −1 (re 2πiθ ),r >1}. 

Let c ∈ M be a parameter such that a strictly pre-periodic dynamical ray Rc(θ) lands at c. Wewant 

to show that the parameter ray RM (θ) lands also at c. 

Set c(t) =Φ −1 (te 2πiθ )fort>1. In other words, we will prove that c(t) → c as t → 1. Denote by 

U = {z : |φc(z)| >t}, U1 = {z : |φ c(t)(z)| >t}, V = {z : |φc(z)| >t 2 } and V1 = {z : |φ c(t)(z)| >t 2 }. 

Then they are Jordan domains. Both Qc : U→Vand Q c(t) : U1 →V1 are proper. The critical value c(t) 

of Q c(t) lies on the boundary of U1 and the post-critical set of Q c(t) is contained in the closure of U1. Set 

ψ = φ −1 

c(t) ◦ φc. Thenψ(U) =U1 and ψ(V) =V1. Moreover, the following diagram commutes: 

U 

ψ 

−→ U1 

Qc ↓ ↓ Q c(t) 

V 

ψ 

−→ V1.


Figure 1 Perturbation F of Qc 

Step 1. Construction of a topological perturbation F of Qc. 

Denote by z(t) =φ−1 c (te2πiθ ). Then z(t) =ψ−1 (c(t)) ∈ ∂U. Letγt = φ−1 c ({re2πiθ , 1 r t}). It is a 

closed arc connecting the point c with the point z(t), whose interior is contained in Rc(θ) ∩ (C \U). 

Let W be a Jordan domain in C \V such that γt ⊂ W . Choose ζ : C → C a homeomorphism that 

is the identity outside W ,withζ(c) =z(t). Set F := ζ ◦ Qc. Then the critical points of F are {0, ∞} 

with F (0) = z(t). Therefore F n (z(t)) = Qn c (z(t)) →∞as n →∞. So PF = {z(t),F(z(t)),...}∪{∞} 

and P ′ ′′ 

F = P F = {∞}. AsFis holomorphic in a neighborhood of the infinity which is a super-attracting 

fixed point of F , we conclude that F is a sub-hyperbolic semi-rational map. 

Lemma 4.1. Thesub-hyperbolicsemi-rationalmapFis c-equivalent to Qc(t). Proof. Let ψ0 be a homeomorphism of C such that ψ0|U = ψ|U. Then ψ0(PF )=PQ and the 

c(t) 

following diagram commutes: 

U 

ψ0 

−→ U1 

F ↓ ↓ Q c(t) 

V 

ψ0 

−→ V1. 

The homeomorphism ψ0 : C \V →C \V1 maps the critical value z(t) ofF to the critical value c(t) of 

Q c(t). Thus there is a unique lift ψ1 : C \U →C \U1 of ψ0, such that ψ1|∂U = ψ|∂U. Obviously, as a lift 

of ψ0, themapψ1 satisfies that ψ1(0) = 0, and the following diagram commutes: 

C \U 

ψ1 

−→ C \U1 

F ↓ ↓ Q c(t) 

C \V 

ψ0 

−→ C \V1. 

Now extend the map ψ1 to a homeomorphism of C by ψ1|U = ψ|U. Then ψ0 ◦ F = Qc(t) ◦ ψ1. Since 

ψ0|U = ψ1|U = ψ|U and PF ⊂ U, by Alexander trick, any homeomorphism of a topological disc which is 

the identity on the boundary is isotopic to the identity rel the boundary. We know that ψ1 and ψ0 are 

isotopic rel PF . This proves that F and Qc(t) are c-equivalent. 

Step 2. Application of the Thurston algorithm to F . 

Let η0 be a Möbius transformation of C such that it is normalized by mapping the triple (0,F2 (z(t)), ∞) 

to the triple (0,Q2 c(t) (c(t)), ∞). As in Section 3, there is a unique homeomorphism η1 of C with the same 

normalization, such that f1 := η0 ◦ F ◦ η −1 

1 is holomorphic on the Riemann sphere C. Since F is 

holomorphic except on F −1 (W )=Q−1 c (W ), we see that η1 is holomorphic except on F −1 (W ).


Recursively, there is a unique homeomorphism ηn of C with the same normalization, such that fn := 

ηn−1 ◦ F ◦ η−1 n is holomorphic on C. Moreoverηnis holomorphic except on n k=1 F −k (W ). Noticing that 

F = Qc except on F −1 (W )=Q−1 c (W ), we have 

n 

F −k (W )= 

k=1 

n 

k=1 

Q −k 

c (W ). 

By Theorem 3.2 and Lemma 4.1, we see that the sequence {fn} converges uniformly to the quadratic 

polynomial Qc(t) and the sequence {ηn} uniformly converges to a continuous map η of C. Notethatηnis holomorphic in V and is normalized by mapping the triple (0,F2 (z(t)), ∞) to the triple (0,Q2 c(t) (c(t)), ∞) 

for all n 0. Thus η is univalent on V and mapping the pair (F 2 (z(t)), ∞) to the pair (Q2 c(t) (c(t)), ∞) 

(note that F 2 (z(t)) = Q2 c (z(t)) ∈V). In particular, η−1 ◦ Qc(t) ◦ η = F = Qc on V. Therefore η|V = ψ|V. 

This is because that the holomorphic conjugation between two super-attracting fixed points of degree 

two is unique. 

ProofofTheorem1.2. The critical point zero is pre-periodic for Qc. SetX0to be the unique periodic 

cycle in PQc. We define Xk and X as in Section 1. 

Note that E := {z : |z| > 8}∪{∞}is compactly contained in V for t ∈ (1, √ 2). For each y ∈ X0 we 

choose a simply connected neighborhood Uy such that Uy ∩ E = ∅ and Uy \{y} is disjoint from PQc. 

Define Ux and Ux(r) for any point x ∈ X and any constant r 0and 

a positive function C(r) defined on r ∈ (0,r0) withC(r) → 0(asr→0) such that for any t>1with 

r(t) 1 such that r(t) 0 such that 

 

|c − c(t)| C0 D(η, E) C0 C(r(t)). 

Therefore |c − c(t)| →0ast → 1sinceC(r) → 0asr → 0andr(t) → 0ast → 1. Now the theorem is 

proved. 

Acknowledgements The first author was supported by National Natural Science Foundation of China (Grant 

Nos. 10831004, 10721061), and by Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences. 

The second author was supported by the EU Research Training Network on Conformal Structures and Dynamics. 

The authors cordially thank the referees for their careful reading and helpful comments. 

References 

1 Beardon A, Gehring F W. Schwarzian derivatives, the Poincare metric and the kernel function. Comment Math Helv, 

1980, 55: 50–64 

2 Beardon A, Pommerenke C. The Poincaré metric of plane domains. J London Math Soc, 1978, 18: 475–483


3 Cui G Z, Tan L. A priori bound for the distortion of univalent functions. Preprint 

4 Douady A, Hubbard J H. Etude dynamique des polynômes complexes (première partie). Publications Mathématiqus 

d’Orsay, 84-02, 1984 

5 Lehto O. Univalent Functions and Teichmüller Spaces. Berlin: Springer-Verlag, 1987 

6 McMullen C. Complex Dynamics and Renormalizations. Annals of Mathematics Studies 135. Princeton: Princeton 

University Press, 1994 

7 Shishikura M. On a theorem of Mary Rees for matings of polynomials. In: Tan L, ed. The Mandelbrot Set, Theme 

and Variations. LMS Lecture Note Series 274. Cambridge: Cambridge University Press, 2000, 289–305

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