A GENERALIZATION OF DOUADY’S FORMULA

. The Douady’s formula was deﬁned for the external argument on the boundary points of the main hyperbolic component W 0 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.

For θ ∈ T = R/Z, the external ray of argument θ of the Mandelbrot set is the curve 1} . An external ray is said to land at c if lim r→1 Φ −1 M (re 2πiθ ) = c. In this case, we say that θ is an external argument of c.
A quadratic polynomial P c is called hyperbolic if it has an attracting cycle. A component W of the interior of M is hyperbolic if P c is hyperbolic for some c ∈ W. In fact, in this case P c 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 , W 0 , is defined as the set of parameters c ∈ C for which P c has an attracting fixed point α c . The boundary of W 0 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 P c has an attracting cycle of period k for every c ∈ W. The map ρ W : W → D, defined by the derivative of P k c at the attracting periodic point, is an analytic isomorphism which can be extended continuously to the boundary of W. Using ρ W , 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 internal argument zero is called the root of W. If c = 1 4 is a parabolic parameter, 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 W 0 with internal argument γ and external arguments θ − , θ + , with 0 < θ − < θ + < 1 3 , then T (θ − ) is an external argument of a real Misiurewicz parameter and T (θ + ) is an external argument of c ∈ 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 P c , 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.
As in the main hyperbolic component W 0 , k = 1, a 1 = 0 and b 1 = 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 c 0 the center of W. There is a copy of M W inside of M , in which W corresponds to the main cardioid W 0 . More precisely, there is a continuous bijection can be obtained by taking in K c0 a component U and replacing U by a copy of K c . In particular, it is known the following result [3].
. is an external argument of c ∈ M, then the corresponding external argument θ of c 0 ⊥ c is given by the following algorithm: then the external arguments (θ + t , θ − t ) at c t can be obtained by tuning. Explicitly, where θ − 0 and θ + 0 are the external arguments in the parameterĉ ∈ ∂W 0 with internal argument t.
In fact, for every external angle θ at ∂W there is an external angle θ 0 at the main cardioid. Moreover, the map θ → θ 0 is monotone, one to one and preserves the type of the landing point.
Lemma 2.2. If θ ± and θ ± 0 , are as above, then Proof. Since P ct has a parabolic periodic point, with period kq, the external arguments in the parabolic parametersĉ ∈ ∂W 0 and c t ∈ ∂W, can be written as . . t ± kq , respectively. By definition we have, 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.  is [a 1 , a 2 , a 3 , ...] where a = a 0 + 1 For each n, the truncated continued fraction [a 0 , a 1 , ..., a n ] is a rational number p n /q n known as the convergent of a. Let U and V be two domains with U compactly contained in V , in notation U 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 f c is called renormalizable if there is an iterate n, two neighborhoods U and V around 0 satisfying U V , such that the restriction of f n c 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 renormalization. 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 f c .
When c is a real parameter, a central cascade of f c is a sequence U m of neighborhoods of 0 such that U m+1 U m and the first return of 0 to U m belongs to U m+1 . The first time n m such that f nm c (0) ∈ U m is called the m-central return of f c . For 0 < α ≤ 1, a quadratic map f c is said to satisfy a summability condition of order α if the series 1 |(f n c ) (c)| α 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 θ ∈ A, then the ray T (θ) lands at the real line at some parameter c . Consider the set RF of all real parameters c such that c 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 q 2n+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 is of bounded type, then the map f c satisfies a summability condition of order 1 2 (see Lemma 5.4 in [1] and the proof of Proposition 3.1 in [8]). We call a parameter c ∈ 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. 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 g m is quasiconformally conjugated to f c . This implies that the moduli of the central returns of g are comparable with the moduli v n of the central returns of f c (for more details see [7]). In Lemma 5.4 of [1] it is shown that the moduli satisfy the Martens-Nowicki's condition, [8]. By quasiconformality, if v k are the moduli of the central returns of g we have As in M. Martens and T. Nowicki, the quotient | v n v n+1 | is a lower bound of |(g n ) (c)|. Hence g satisfies a summability condition with exponent 1 2 , (see the proof of Proposition 3.1 in [8]).
The previous lemma has the following consequence: If θ is an irrational external argument in the boundary of W between the external arguments θ − and .a 1 . . . a k b 1 b 2 . . . b k then the external ray with angle F (θ) lands in a parameter c ∈ M which is finitely renormalizable. Furthermore, the corresponding Julia set J c is locally connected and the map f c admits an absolutely continuous invariant measure with respect to Lebesgue.
Proof. The map f c renormalizes to a polynomial with parameter c in RF which is non-renormalizable. Then, in fact, f c is only 1-renormalizable. By Yoccoz's Theorem M is locally connected at c and is the landing point of at least one ray. By hypothesis, the angle θ of the ray landing at c 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 f c 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 < continuous invariant measure [5]. Here µ max is maximal multiplicity of the critical points, which for this quadratic polynomial is equal to 1.