Distortion control of conjugacies between quadratic polynomials

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1 SCIENCE CHINA Mathematics. ARTICLES. March 2010 Vol. 53 No. 3: doi: /s 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 , China; 2 Laboratoire de Mathématiques, UMR du CNRS 6093, Université d Angers, 2 bd. Lavoisier, Angers cedex 01, France 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 MSC(2000): distortion, conjugacy, polynomial 37F10, 30C35 Citation: Cui G Z, Tan L. Distortion control of conjugacies between quadratic polynomials. Sci China Math, 2010, 53(3): , doi: /s 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 S f, 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

2 626 CUI GuiZhen et al. Sci China Math March 2010 Vol. 53 No A priori bound Let g be a geometrically finite rational map, i.e., the post-critical set P g (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 X 0 be the union of finitely many periodic cycles contained in the Julia set of g. It is known that every point in X 0 is either parabolic or repelling. Set X 1 = g 1 (X 0 )\X 0, X n+1 = g n (X 1 )andx = n 0 X n. Now for each y X 0,chooseU y a simply connected neighborhood of y satisfying the following properties: (1) These domains U y are disjoint pairwise. (2) For each n 1andeachpointx X n with g n (x) =y X 0,letU x be the connected component of g n (U y ) containing x, and then the map g n : U x \{x} U y \{y} is a covering (this implies that U x P g is either empty or equal to {x}). Denote by D the unit disc. For any point y X 0, there is a Riemann mapping χ from U y to D with χ(y) =0. SetU y (r) =χ 1 ({z : z <r}). For any point x X n,setu x (r) to be the component of g n (U g n (x)(r)) that contains the point x. Refer to [3] for the next theorem. Theorem 1.1 (A priori bound). Let (g, {U x } x X ) be a pair defined as above. Let E be an open set compactly contained in C \ X. Set n V n (r) =C k=0 x X k U x (r) for n 0. Then there exist a constant r 0 > 0 and a positive function C(r) defined on (0,r 0 ) with C(r) 0(as r 0) such that for any n 0 and any univalent holomorphic map φ : V n (r) C, 1.3 External rays D(φ, E) C(r). Now let us consider quadratic polynomials Q c (z) =z 2 + c. DenotebyK c the filled-in Julia set of Q c.for any c C, there exists a conformal map φ c defined on a neighborhood of the infinity such that φ c ( ) =, φ c (z)/z 1asz and φ c Q c (z) =(φ c (z)) 2.Themapφ c is called the Böttcher coordinate of Q c at the infinity. If K c is connected, then φ c defines a Riemann mapping φ c : C \ K c C \ D. Set R c (θ) ={φ 1 c (re 2πiθ ),r >1}. It is called the dynamical θ-ray for Q c. Even if K c is disconnected, φ c 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 : {Q n c (c)} n 0 is bounded}. Equivalently, the point c is contained in M if and only if K c is connected. Define Φ(c) =φ c (c). It turns out that Φ : C\M C\D is a conformal map with Φ(z)/z 1asz (refer to [4]). Set R M (θ) ={Φ 1 (re 2πiθ ),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 R c (θ) lands at c. Then the parameter ray R M (θ) lands also at c. Such a polynomial Q c 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.

3 CUI GuiZhen et al. Sci China Math March 2010 Vol. 53 No 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, E 1 ) D(φ, E) if E 1 E. (d) Assume that φ n : E C is a sequence of univalent functions that converges locally uniformly to a univalent function φ. ThenD(φ, E) lim inf n D(φ n,e). 2.1 Hyperbolic sup-norm of the Schwarzian derivative Let E C be a hyperbolic open set. Denote by ρ E (z) thepoincarédensityofe. Let φ : E C be a univalent function. The Schwarzian derivative of φ is defined by S φ (z) = φ (z) φ (z) 3 2 ( φ ) 2 (z) φ. (z) Theorem 2.2. There is a universal constant C 1 > 0 such that sup S φ (z) ρ E (z) 2 C 1 D(φ, E). (1) z E 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 C 2 > 0 such that sup S φ (z) ρ E (z) 2 C 2. (2) z E 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 C 2 = 3. Here we give an independent proof using only the estimation of PoincarédensityandKöebe 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 densityρ D and f : D C be a univalent function. Then sup z D S f (z) ρ 2 D (z) 3/2. (4) We say that two positive quantities ρ 1 and ρ 2 are comparable (denoted by ρ 1 ρ 2 ) if there is a universal constant C>1 such that 1/C < ρ 1 /ρ 2 <C. (C) (Beardon-Pommerenke theorem, refer to Theorem 1 in [2] or Theorem 2.3 in [6]) Let U be a hyperbolic open set contained in C with Poincaré densityρ U (z), and then 1/ρ U (z) d(z, U)(1 + mod(z,u)), (5) where d(z, U) is the Euclidean distance from z to the boundary of U and mod(z,u) is the supremum of the moduli of essential round annuli in U whose core curves pass through the point z (if no such annulus exists, set mod(z,u) = 0). Denote by ρ A(R) the Poincaré density of the annulus A(R) ={z :1/R < z <R}. Lemma 2.5. For any univalent function ψ : A(R) C with R>32, we have ( 2 S ψ (1) ρ 2 12 ) 2 A(R) (1) log R. π(r 32)

4 628 CUI GuiZhen et al. Sci China Math March 2010 Vol. 53 No. 3 Proof. Fix a constant r with 32 <r<r. Then there is a quasiconformal map h of C such that h = ψ on the annulus A(r) ={z :1/r < z <r}. By post-composing with a Möbius transformation, we may assume that h is normalized by fixing 0, 1and. Note that the Schwartzian derivative S ψ (1) remains unchanged after composing a Möbius transformation. Let h 1 be the normalized quasiconformal map of C with Beltrami coefficient μ h1 = μ h in D(r) ={z : z <r} and μ h1 = 0 otherwise. Then h 1 is holomorphic in Δ(1/r) ={z : z > 1/r}. Seth 2 = h h 1 1. It is holomorphic in h 1 (D(r)). By the composition formula of the Schwarzian derivative, we have S ψ (1) = S h2 h 1 (1) = S h1 (1) + S h2 (1)(h 1(1)) 2. Since h 1 is univalent in Δ(1/r), where the Poincaré densityis2r/(r 2 z 2 1), by (4) we have S h1 (1) (r2 1) 2 4r (6) Since h 1 ({1 < z <r}) separates {0, 1} from h 1 ({ z = r}), the Euclidean distance between the point 1andh 1 ({ z = r}) isatleastr 1 := 2 6 (r 32) by (3). So h 2 is univalent on the disc D(1,r 1 )={z : z 1 <r 1 }, where the Poincaré densityis2r 1 /(r1 2 z 1 2 ). Thus (r 32)2 S h2 (1) (7) 2 by (4). Now applying Koebe 1/4-theorem to h 1 on D(1, 1 1/r), and noticing that its image avoids the origin, we have h 1 (1) 4r/(r 1). Now let r R, and then we have ( S ψ (1) 1 (R +1) (R 32) (R 32) ) 3 4R 2 2 (R 1) 2 Note that ρ 2 A(R) (1) = 4(log R)2 /(π 2 ). The lemma is proved. ProofofLemma2.4. For any point z E, denotebyd E the disc centered at z with radius d(z, E), and by A an essential round annulus in E with the maximal modulus whose core curve passes through z (if such annulus exists). Then 1/ρ E (z) d(z, E)(1 + mod(a)) by (5). Case 1. mod(a) log 64/π. In this case ρ 1 E (z) d(z, E) ρ 1 D (z), and φ is univalent in D. So S φ (z) ρ E (z) 2 S φ(z) ρ D (z) Case 2. mod(a) > log 64/π. Weproveatfirstthatρ E (z) is comparable to ρ A (z). Applying a linear map to E if necessary, we may assume z =1andA = A(R) forsomer>1. As log R π =mod(a) =mod(z,u) > 6log2, π we have actually R>64. Thus 1/64 d(1, E) 1. From (5), ρ E (z) is comparable to 1/mod(A) which is 2ρ A (z). Thus ρ E (z) is comparable to ρ A (z). By Lemma 2.5 we have S φ (z) ρ E (z) 2 S ( φ(z) 2 12 ) 2 ( ρ A (z) 2 log R 2 12 ) 2 log 64. π(r 32) 32π This ends the proof. Proof of Theorem 2.2. Assume that D(φ, E) =δ<. By Lemma 2.4 the left-hand side of (1) is bounded by a universal constant. So we only need to consider the case when δ is small. For example we may assume that δ<1/64. The symbols C below denote some universal constants.

5 CUI GuiZhen et al. Sci China Math March 2010 Vol. 53 No For any z E, choosez 1 E such that ρ E1 (z) 2ρ E (z) fore 1 := E \{z 1 }.ChoosenowtwoMöbius transformations β and γ such that β(z 1 )=, β(z) =1,d(1, β(e 1 )) = 2 such that ψ := γ φ β 1 fixes 0, 1and. SetG := β(e 1 ). It is contained in C. Wehave S φ (z) ρ 2 E (z) 4 S φ(z) ρ 2 E 1 (z) =4 S ψ(1). (8) (1) Since δ<1/64, we have δ(1 + log δ ) 2 2 δ. Thus in order to prove (1), we only need to show that S ψ (1) ρ 2 G (1) C 3δ(1 + log δ ) 2. (9) Case A. mod(1,g) log δ /π. By our choice of G, wehaved(1, G) = 2. By (5), ρ G (1) 2 (1+mod(1,G)) 2 (1+ log δ ) 2. Applying Lemma 2.3 to ψ, weget log ψ (z) 5πδ for z G. Thisimpliesthatψ maps {z : z 1 < 2} into A δ := {e 6πδ w e 6πδ }. Therefore ψ ({z : z 1 1}) is contained in B δ := {z A δ : d p (z,ψ (1)) d 0 (0, 1/2)} by Ahlfors-Schwarz lemma, where d p is the Poincaré metricona δ,andd 0 is the Poincaré metric on the unit disc. One can then check easily that the Euclidean diameter of B δ is less than C 4 δ for a universal constant C 4 > 0. Thus ψ (z) ψ (1) C 4 δ 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) C 5 δ. Combining with ψ (1) A δ, we get S ψ (1) C 6 δ. Therefore S ψ (1) ρ G (1) 2 C 7 δ(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 S ψ (1) Case 2 ρ 2 G (1) Hence in both cases (9), (1) is proved. 2.2 Controlling distortions of conjugations ρ 2 G log R/π =mod(a) =mod(1,g) > log δ /π. S ( ψ(1) 2 12 ) 2 ( log R 2 12 ) 2 log δ ρ 2 <C 8 δ(1 + log δ ) 2. A (1) π(r 32) π(1/δ 32) Denote Q c (z) =z 2 + c. Denote by φ c the Böttcher coordinate of Q c at the infinity, M the Mandelbrot set, and Φ(c) =φ c (c). Then Φ : C \ M C \ D is a conformal map. Note that if Φ(c) 2, then φ 1 c is univalent in {z : z > 2} and hence φ c is univalent in E = {z : z > 8} { } by Koebe distortion theorem. Lemma 2.6. There is a universal constant C 0 > 0, such that for any points c 1,c 2 C \ M with Φ(c i ) 2(i =1, 2), we have c 1 c 2 C 0 D(ψ, E), where ψ = φ 1 c 2 φ c1. Proof. Let φ c (z) =z+b 0 +b 1 /z+ be the expansion at the infinity. By the formula φ c Q c (z) =φ c (z) 2, one may check that b 0 =0andb 1 = c/2. Therefore ψ(z) has the expansion ψ(z) =z +(c 1 c 2 )/(2z)+ at the infinity. Let Ψ(z) =1/ψ(1/z) =z + a 2 z 2 + a 3 z 3 +. Using ψ(1/z)ψ(z) 1 one obtains that a 2 =0and a 3 =(c 2 c 1 )/2. This implies that S Ψ (0) = 6a 3 =3(c 2 c 1 ). 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 S ψ (z) lim z ρ 2 (z) = S Ψ(0) ρ 2 (0) = 3(c 2 c 1 ). 256

6 630 CUI GuiZhen et al. Sci China Math March 2010 Vol. 53 No. 3 Therefore by Theorem 2.2, c 1 c 2 = S ψ (z) lim z ρ 2 C 0 D(ψ, E). (z) 3 Thurston algorithm 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 P F is defined to be the closure of the forward orbits of the critical points. Denote by P F the accumulation set of P F. 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 F 1 and F 2 are c-equivalent if there is a pair of homeomorphisms (φ 0,φ 1 )ofc such that (a) φ 0 F 1 = F 2 φ 1 ; (b) the two maps φ 0 and φ 1 are isotopic rel P F1 ;and(c) φ 0 is holomorphic in a neighborhood of P F 1 (hence φ 1 is also holomorphic and coincides with φ 0 in a neighborhood of P F 1 ). 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 P F which meet critical points. Assume that F is holomorphic in a neighborhood of P F. Pick three distinct points in P F. 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 (φ 0,φ 1 ). Since F is holomorphic in a neighborhood of P F,thereisa pair of normalized homeomorphisms (ξ 0,ξ 1 )ofc in the isotopy class of φ 0 rel P F, such that they are holomorphic and coincide with each other in a neighborhood of P F,andξ 0 F = f ξ 1. Furthermore, for n 2, there is a normalized homeomorphism ξ n of C in the isotopy class of ξ 0 rel P F such that ξ n 1 F = f ξ n and ξ n coincides with ξ 0 in a neighborhood of P F. Lemma 3.1. The sequence {ξ n } is uniformly convergent. This lemma is proved by Shishikura and Rees in the case that P F 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 η 0 is a normalized homeomorphism of C such that η 0 is holomorphic in a neighborhood of P F. Then there is a unique normalized homeomorphism η 1 of C such that f 1 := η 0 F η1 1 is holomorphic. Obviously, η 1 is also holomorphic in a neighborhood of P F. Recursively, there is a unique normalized homeomorphism η n of C such that f n := η n 1 F ηn 1 is holomorphic. Then η n is also holomorphic in a neighborhood of P F. The sequence of rational maps {f n} 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 {f n } converges uniformly to the rational map f. Moreover, the sequence {η n } is also uniformly convergent. Proof. By the definition we know that (a) φ 0 F = f φ 1 ; (b) the two maps φ 0 and φ 1 are isotopic rel P F ;and(c)bothφ 0 and φ 1 are holomorphic in a neighborhood of P F. Since F is holomorphic in a neighborhood of P F,wemaymodifyφ 0 in its homotopy class such that both φ 0 and φ 1 are holomorphic and coincide in a neighborhood of P F. As above, we have a sequence of normalized homeomorphisms

7 CUI GuiZhen et al. Sci China Math March 2010 Vol. 53 No {φ n } n 1 of C in the isotopy class of φ 0 rel P F, such that φ n 1 F = f φ n and φ n = φ 0 in a neighborhood of P F. Set ψ n = η n φ 1 n.thenψ n is holomorphic in a neighborhood of P f. We may assume furthermore that ψ 0 is a K-quasiconformal map of C by modifying φ 0 in its isotopy class. Then ψ n is also K-quasiconformal. It is easy to check that ψ n 1 f = f n ψ n. See the following diagram: C f C C f C f C φ n φ n 1 φ 2 φ 1 φ 0 C F C C F C F C η n η n 1 η 2 η 1 η 0 C f n C C f 2 C f 1 C. Since ψ 0 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 ψ n 1 f = f n ψ n, we see that ψ n is also a K-quasiconformal map of C and holomorphic in f n (W ). Therefore there is a subsequence of {ψ n } 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 {ψ n } converges uniformly to the identity. It follows that the Thurston sequence {f n } converges uniformly to the rational map f. Because η n = ψ n φ n, from Lemma 3.1, we know that the sequence {φ n }, and therefore the sequence {η n } is uniformly convergent. 4 Misiurewicz-hyperbolic deformation We now begin to prove Theorem 1.2. Recall that φ c is the Böttcher coordinate of the quadratic polynomial Q c (z) =z 2 + c at infinity. For c M, themapφ c defines a Riemann representation φ c : C \ K c C \ D. The dynamical θ-ray is defined by R c (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) =φ c (c). The parameter θ-ray is defined by R M (θ) ={Φ 1 (re 2πiθ ),r >1}. Let c M be a parameter such that a strictly pre-periodic dynamical ray R c (θ) lands at c. Wewant to show that the parameter ray R M (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 : φ c (z) >t}, U 1 = {z : φ c(t) (z) >t}, V = {z : φ c (z) >t 2 } and V 1 = {z : φ c(t) (z) >t 2 }. Then they are Jordan domains. Both Q c : U Vand Q c(t) : U 1 V 1 are proper. The critical value c(t) of Q c(t) lies on the boundary of U 1 and the post-critical set of Q c(t) is contained in the closure of U 1.Set ψ = φ 1 c(t) φ c.thenψ(u) =U 1 and ψ(v) =V 1. Moreover, the following diagram commutes: U Q c V ψ U 1 Q c(t) ψ V 1.

8 632 CUI GuiZhen et al. Sci China Math March 2010 Vol. 53 No. 3 Figure 1 Perturbation F of Q c Step 1. Construction of a topological perturbation F of Q c. Denote by z(t) =φ 1 c (te 2πiθ ). Then z(t) =ψ 1 (c(t)) U. Letγ t = φ 1 c ({re 2πiθ, 1 r t}). It is a closed arc connecting the point c with the point z(t), whose interior is contained in R c (θ) (C \U). Let W be a Jordan domain in C \V such that γ t W. Choose ζ : C C a homeomorphism that is the identity outside W,withζ(c) =z(t). Set F := ζ Q c. Then the critical points of F are {0, } with F (0) = z(t). Therefore F n (z(t)) = Q n c (z(t)) as n. So P F = {z(t),f(z(t)),...} { } and P F = P F = { }. AsF is 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-rationalmapF is c-equivalent to Q c(t). Proof. Let ψ 0 be a homeomorphism of C such that ψ 0 U = ψ U. Then ψ 0 (P F )=P Qc(t) and the following diagram commutes: ψ 0 U U1 F Q c(t) V ψ 0 V1. The homeomorphism ψ 0 : C \V C \V 1 maps the critical value z(t) off to the critical value c(t) of Q c(t). Thus there is a unique lift ψ 1 : C \U C \U 1 of ψ 0, such that ψ 1 U = ψ U. Obviously, as a lift of ψ 0,themapψ 1 satisfies that ψ 1 (0) = 0, and the following diagram commutes: C \U F C \V ψ 1 C \U1 Q c(t) ψ 0 C \V1. Now extend the map ψ 1 to a homeomorphism of C by ψ 1 U = ψ U. Then ψ 0 F = Q c(t) ψ 1. Since ψ 0 U = ψ 1 U = ψ U and P F 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 ψ 1 and ψ 0 are isotopic rel P F. This proves that F and Q c(t) are c-equivalent. Step 2. Application of the Thurston algorithm to F. Let η 0 be a Möbius transformation of C such that it is normalized by mapping the triple (0,F 2 (z(t)), ) to the triple (0,Q 2 c(t) (c(t)), ). As in Section 3, there is a unique homeomorphism η 1 of C with the same normalization, such that f 1 := η 0 F η1 1 is holomorphic on the Riemann sphere C. SinceF is holomorphic except on F 1 (W )=Q 1 c (W ), we see that η 1 is holomorphic except on F 1 (W ).

9 CUI GuiZhen et al. Sci China Math March 2010 Vol. 53 No Recursively, there is a unique homeomorphism η n of C with the same normalization, such that f n := η n 1 F ηn 1 is holomorphic on C. Moreoverη n is holomorphic except on n k=1 F k (W ). Noticing that F = Q c 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 {f n } converges uniformly to the quadratic polynomial Q c(t) and the sequence {η n } uniformly converges to a continuous map η of C. Notethatη n is holomorphic in V and is normalized by mapping the triple (0,F 2 (z(t)), ) to the triple (0,Q 2 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 (Q 2 c(t) (c(t)), ) (note that F 2 (z(t)) = Q 2 c (z(t)) V). In particular, η 1 Q c(t) η = F = Q c 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 Q c.setx 0 to be the unique periodic cycle in P Qc. We define X k and X as in Section 1. Note that E := {z : z > 8} { }is compactly contained in V for t (1, 2). For each y X 0 we choose a simply connected neighborhood U y such that U y E = and U y \{y} is disjoint from P Qc. Define U x and U x (r) for any point x X and any constant r<1asinsection 1. In particular, as c X k for some k 1, U c is a neighborhood of c which is a component of Q k c (U y )fory = Q k (c). Recall that γ t = φ 1 c ({re 2πiθ, 1 r t}). Obviously the diameter diamγ t 0ast 1. Thus there are a constant t 0 (1, 2) and a positive function r(t) on(1,t 0 )withr(t) 0ast 1 such that γ t U c (r(t)) for t (1,t 0 ). Recall that the homeomorphism η n defined in Step 2 is holomorphic on the complement of the closure of n k=1 Q k c (W ). For t (1,t 0 ), since γ t U c (r(t)), we may require that the Jordan domain W chosen in Step 1 satisfies that W U c (r(t)). Then the homeomorphism η n is holomorphic on the complement of the closure of n k=1 Q k c (U c (r(t))). Now applying Theorem 1.1 to the pair (Q c, {U x } x X )andthemapη n,wehaveaconstantr 0 > 0and a positive function C(r) defined on r (0,r 0 )withc(r) 0(asr 0) such that for any t>1with r(t) <r 0 and any integer n 1, D(η n,e) C(r(t)). Since r(t) 0ast 1, there exists a constant t 0 > 1 such that r(t) <r 0 for t (1,t 0 ). So we have D(η n,e) C(r(t)) for t (1,t 0 ) and any integer n 1. In particular D(η, E) C(r(t)) for t (1,t 0 ) by Lemma 2.1(d). Recall that η = ψ = φ c(t) φ 1 c on V, therefore on E. By Lemma 2.6, there is a universal constant C 0 > 0 such that c c(t) C 0 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 , ), 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: Beardon A, Pommerenke C. The Poincaré metric of plane domains. J London Math Soc, 1978, 18:

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