A GENERALIZATION of DOUADY's FORMULA Gamaliel Blé Carlos Cabrera (Communicated by Enrique Pujals) 1. Introduction. in This Pa

DISCRETE AND CONTINUOUS doi:10.3934/dcds.2017267 DYNAMICAL SYSTEMS Volume 37, Number 12, December 2017 pp. 6183–6188

A GENERALIZATION OF DOUADY’S FORMULA

Gamaliel Ble´∗ DivisiónAcadémicade Ciencias Básicas,UJAT Km. 1, Carretera Cunduacán–Jalpade Méndez C.P. 86690, CunduacánTabasco, México Carlos Cabrera Instituto de Matemáticasde la UNAM, Unidad Cuernavaca Av. Universidad s/n. Col. Lomas de Chamilpa C.P. 62210, Cuernavaca, Morelos, México

(Communicated by Enrique Pujals)

Abstract. The Douady’s formula was deﬁned for the external argument on the boundary points of the main hyperbolic component W0 of the Mandelbrot set M and it is given by the map T (θ) = 1/2 + θ/4. We extend this formula to the boundary of all hyperbolic components of M and we give a characterization of the parameter in M with these external arguments.

1. Introduction. In this paper we will restrict our attention to the dynamics of 2 complex quadratic polynomials Pc(z) = z +c. In this case, the dynamical plane can be decomposed into two complementary sets: The filled Julia set Kc which consists of all points with bounded orbit and the basin of infinity Ac(∞). The boundary of Kc is called the Julia set Jc. ˆ When we consider the polynomial Pc in the extended complex plane C = C ∪ {∞}, the point at infinity is a super-attracting fixed point and hence there exists a neighborhood of infinity Uc and an analytic isomorphism, known as the Böttcher 0 map, φc : Uc → {z ∈ C : |z| > R}, such that φc(∞) = ∞, φc(∞) = 1 and 2 φc ◦Pc = [φc] . If the critical point 0 is in Kc, then Kc is connected and Uc = C\Kc, otherwise Uc is a neighborhood of infinity, containing the critical value c, see for instance [4]. The Mandelbrot set M is defined as the set

M = {c ∈ C : Kc is connected}. ˆ ˆ A. Douady and J.H. Hubbard deﬁned the map ΦM : C \ M → C \ D, given by ΦM (c) = φc(c) and they proved that ΦM is an analytic isomorphism satisfying 0 ΦM (∞) = ∞ and ΦM (∞) = 1, [2, 4].

2010 Mathematics Subject Classification. Primary: 37F10, 37F45; Secondary: 37F50. Key words and phrases. Mandelbrot set, quadratic polynomials, tuning map, summability condition, absolutely continuous invariant measures. The authors are grateful to CONACYT for financial support CB-2012/181247 and CB- 2015/255633 given to this work. ∗ Corresponding author: Gamaliel Blé.

6183 6184 GAMALIEL BLE´ AND CARLOS CABRERA

For θ ∈ T = R/Z, the external ray of argument θ of the Mandelbrot set is the curve −1 2πiθ RM (θ) = ΦM {re : r > 1} . −1 2πiθ An external ray is said to land at c if limr→1 ΦM (re ) = c. In this case, we say that θ is an external argument of c. A quadratic polynomial Pc is called hyperbolic if it has an attracting cycle. A component W of the interior of M is hyperbolic if Pc is hyperbolic for some c ∈ W. In fact, in this case Pc is hyperbolic for all c ∈ W and it is conjectured that all components of the interior of M are hyperbolic [2, 4]. The main hyperbolic component of M, W0, is defined as the set of parameters c ∈ C for which Pc has an attracting fixed point αc. The boundary of W0 is called the main cardioid of M. It is known that for every hyperbolic component W of the interior of M there exists k ∈ N fixed, such that Pc has an attracting cycle of k period k for every c ∈ W. The map ρW : W → D, defined by the derivative of Pc at the attracting periodic point, is an analytic isomorphism which can be extended continuously to the boundary of W. Using ρW , we can define the internal argument γ for all c in the boundary of W , [2]. On the other hand, from Yoccoz’s Theorem, M is locally connected for all parameters in the boundary of hyperbolic components [6]. Hence, a parameter c ∈ M in the boundary of a hyperbolic component W has well defined internal and external arguments, [9, 4]. The parameter c ∈ ∂W with 1 internal argument zero is called the root of W. If c 6= is a parabolic parameter, 4 then c has a rational internal argument and two external arguments θ− < θ+, [4]. Douady gives a formula that relates the parameters in the main cardioid with real parameters in M. The map induced by this formula takes a parameter with external argument θ and sends it onto a real parameter with external argument T (θ) = 1/2 + θ/4. In [1], the following was proved.

Theorem 1.1. If c is a parabolic point of the boundary of W0 with internal ar- − + − + 1 − gument γ and external arguments θ , θ , with 0 < θ < θ < 3 , then T (θ ) is an external argument of a real Misiurewicz parameter and T (θ+) is an external argument of c0 ∈ M ∩ R, the root of a primitive hyperbolic component. Furthermore, if γ is irrational and satisﬁes an asymmetrical Diophantine condition then there exists an absolutely continuous invariant measure for Pc0 , see [1]. In this work, we extend this formula to the boundary of all hyperbolic components and we give a characterization of the parameter with these external arguments.

2. Generalization of Douady’s formula. It is known that if W ⊂ M is a hyperbolic component of the interior of M then the root c ∈ W has two external 1 arguments θ− and θ+ , with the exception when c = , [4]. Hence we can associate W W 4 − + to W the couple (θW , θW ). Let W be a given hyperbolic component of the interior of M, with period k and − + (θ = .a1a2 . . . ak, θ = .b1b2 . . . bk), the external rays in the root of W. If θ is an external argument in the boundary of W we deﬁne the map θ b b a a F (θ) = + 1 + ... + k + 1 + ... k 4k 2 2k 2k+1 22k = .b1b2 . . . bka1a2 . . . akθ. A GENERALIZATION OF DOUADY’S FORMULA 6185

As in the main hyperbolic component W0, k = 1, a1 = 0 and b1 = 1, this map generalizes the Douady’s formula and we will show that it has similar properties of the map T . Before, we give some basic concepts and properties of tuning that can be found in [3]. Let W be a hyperbolic component of M, of period k, and c0 the center of W. There is a copy of MW inside of M, in which W corresponds to the main cardioid W0. More precisely, there is a continuous bijection ψW : M → MW , such that ψW (0) = c0, ψW (W0) = W and ∂MW ⊂ ∂M. For all c in M the point ψW (c) is called c0 tuned by c and it is denoted by c0 ⊥ c or W ⊥ c. The ﬁlled Julia set Kc0⊥c can be obtained by taking in Kc0 a component U and replacing U by a copy of Kc. In particular, it is known the following result [3]. Theorem 2.1. (Douady-85) Let W be a hyperbolic component of the interior of − + M with period k and (θ = .a1a2 . . . ak, θ = .b1b2 . . . bk) be the external rays in the root c1 of W. If θ = .s1s2s3 ... is an external argument of c ∈ M, then the 0 corresponding external argument θ of c0 ⊥ c is given by the following algorithm: 0 − + 0 0 0 θ = (θ , θ ) ⊥ θ = .s1s2s3 ..., 0 0 where si = a1a2 . . . ak if si = 0 and si = b1b2 . . . bk if si = 1. From now on, we suppose that W is a hyperbolic component of the interior of − + M with period k and (θ = .a1a2 . . . ak, θ = .b1b2 . . . bk) are the external rays in the root of W and θ− < θ+. p Remark 1. If c ∈ ∂W has a rational internal argument t = , with (p, q) = 1, t q + − then the external arguments (θt , θt ) at ct can be obtained by tuning. Explicitly, ± − + ± θt = (θ , θ ) ⊥ θ0 , − + where θ0 and θ0 are the external arguments in the parameterc ˆ ∈ ∂W0 with internal argument t.

In fact, for every external angle θ at ∂W there is an external angle θ0 at the main cardioid. Moreover, the map θ 7→ θ0 is monotone, one to one and preserves the type of the landing point.

± ± Lemma 2.2. If θ and θ0 , are as above, then ± − + ± F (θt ) = (θ , θ ) ⊥ T (θ0 ).

Proof. Since Pct has a parabolic periodic point, with period kq, the external arguments in the parabolic parametersc ˆ ∈ ∂W0 and ct ∈ ∂W, can be written as ± ± ± ± ± ± ± ± θ0 = .s1 s2 . . . sq and θt = .t1 t2 . . . tkq, respectively. By deﬁnition we have,

± ± ± ± T (θ0 ) = .10s1 s2 . . . sq , and ± ± ± ± F (θt ) = .b1b2 . . . bka1a2 . . . akt1 t2 . . . tkq. From Remark 1, we have ± − + ± − + ± F (θt ) = F ((θ , θ ) ⊥ θ0 ) = (θ , θ ) ⊥ T (θ0 ). 6186 GAMALIEL BLE´ AND CARLOS CABRERA

The previous lemma can be generalized to all external angles whose rays land at ∂W . Hence, we have that the generalized Douady’s formula is nothing but the tuning of the original formula. By the Theorem 1.1, Theorem 2.1, Lemma 2.2 and the deﬁnition of F (θ) we obtain the following result. Theorem 2.3. Let W be a hyperbolic component of the interior of M with period k − + and (θ = .a1a2 . . . ak, θ = .b1b2 . . . bk) be the external rays at the root of W. If θ is − an external argument in the boundary of W and θ < θ < .a1 . . . akb1b2 . . . bk. Then − + F (θ) is an external argument of c ∈ M. In particular, if θc , θc are the external arguments in a parabolic point c ∈ W, then + 1. F (θc ) is an external argument of the root of a primitive hyperbolic component. − 2. F (θc ) is an external argument of a Misiurewicz parameter. Let W be a hyperbolic component. The interval [−2, 0] tuned with W is a curve in M that we call the main vein of the little Mandelbrot copy starting at c0. + − Corollary 1. The external ray with angles F (θc ) or F (θc ) land in the main vein of the hyperbolic component W .

3. Summability condition. Given a real number a, the continued fraction expansion of a is [a1, a2, a3, ...] where 1 a = a + 0 1 a + 1 1 a2 + . a3 + ···

For each n, the truncated continued fraction [a0, a1, ..., an] is a rational number pn/qn known as the convergent of a. Let U and V be two domains with U compactly contained in V , in notation U b V .A quadratic-like map g : U → V is a degree 2 branched covering. Given a quadratic-like map, the little filled Julia set is the set of points which can be iterated infinitely many times. The map fc is called renormalizable if there is an iterate n, two neighborhoods U and V around 0 satisfying U b V , such that the restriction n of fc to U is a quadratic-like map with connected little filled Julia set. In fact, the operation of tuning can be seen as the inverse operation to renor- malization. For a hyperbolic component W , the map W ⊥ c is renormalizable with iterate equal to the period of W , and the induced quadratic-like map is quasiconformally conjugated to fc. When c is a real parameter, a central cascade of fc is a sequence Um of neighborhoods of 0 such that Um+1 b Um and the first return of 0 to Um belongs to Um+1. nm The first time nm such that fc (0) ∈ Um is called the m-central return of fc. For 0 < α ≤ 1, a quadratic map fc is said to satisfy a summability condition of order α if the series X 1 n 0 α |(fc ) (c)| converges (see [10]). Let A be the set of irrational angles θ and such that the external ray with angle θ lands at the main cardioid. By Douady-Hubbard and Yoccoz’s theorems, the set landing points of rays with angles in A consists precisely of the parameters c with irrational internal angle (see [4] and [6]). In [1], the first author showed that if A GENERALIZATION OF DOUADY’S FORMULA 6187

θ ∈ A, then the ray T (θ) lands at the real line at some parameter c0. Consider the set RF of all real parameters c0 such that c0 is the landing point of a ray with angle T (θ) with θ ∈ A. In [1] the first author showed that if c ∈ RF then there is a central cascade around the critical point where the n-central return is equal to q2n+1, the n-convergent of γ, where γ is the internal argument of c. (see Theorem 1.3 (iii) and Lemma 5.2 in [1]). Moreover, if the continuous fraction expansion of γ 1 is of bounded type, then the map f 0 satisfies a summability condition of order c 2 (see Lemma 5.4 in [1] and the proof of Proposition 3.1 in [8]). We call a parameter c0 ∈ RF of bounded type whenever the associated parameter c in the main cardioid has an internal address of bounded type. By making use of the properties of tuning we show that the generalized formula F also has these properties. Lemma 3.1. For c ∈ RF of bounded type, let W be a hyperbolic component of the Mandelbrot set, then W ⊥ c satisfies a summability condition of order 1/2. Proof. Let g = W ⊥ c. By construction g is renormalizable of the same period m of the component W. There exist a neighborhood U around 0 such that the map gm is quasiconformally conjugated to fc. This implies that the moduli of the central returns of g are comparable with the moduli vn of the central returns of fc (for more details see [7]). In Lemma 5.4 of [1] it is shown that the moduli satisfy ∞ 1/2 X vk+1 < ∞, vk k=1 0 the Martens-Nowicki’s condition, [8]. By quasiconformality, if vk are the moduli of the central returns of g we have

∞ 1/2 ∞ 0 1/2 ∞ 1/2 1 X vk+1 X vk+1 X vk+1 ≤ 0 ≤ K < ∞. K vk v vk k=1 k=1 k k=1 0 vn As in M. Martens and T. Nowicki, the quotient | 0 | is a lower bound of vn+1 n 0 1 |(g ) (c)|. Hence g satisﬁes a summability condition with exponent 2 , (see the proof of Proposition 3.1 in [8]). The previous lemma has the following consequence: Theorem 3.2. If θ is an irrational external argument in the boundary of W between − the external arguments θ and .a1 . . . akb1b2 . . . bk then the external ray with angle F (θ) lands in a parameter c0 ∈ M which is ﬁnitely renormalizable. Furthermore, the corresponding Julia set Jc0 is locally connected and the map fc0 admits an absolutely continuous invariant measure with respect to Lebesgue.

Proof. The map fc0 renormalizes to a polynomial with parameter c in RF which is non-renormalizable. Then, in fact, fc0 is only 1-renormalizable. By Yoccoz’s Theorem M is locally connected at c0 and is the landing point of at least one ray. By hypothesis, the angle θ0 of the ray landing at c0 is the tuning of W with the ray ˜ Rθ˜ landing at parameterc ˜ in the boundary of the main cardioid. Hence T (θ) ∈ A ˜ and F (θ) = W ⊥ T (θ). By Lemma 3.1, the map fc0 satisﬁes a summability condition with exponent 1/2. J. Graczyk and S. Smirnov showed that when a map satisﬁes a summability condition with exponent α < 2 then admits an absolutely 2+µmax 6188 GAMALIEL BLE´ AND CARLOS CABRERA continuous invariant measure [5]. Here µmax is maximal multiplicity of the critical points, which for this quadratic polynomial is equal to 1.

Acknowlegments. We would like to thank the referee for helpful comments to improve this manuscript.

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