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

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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´∗ Divisi´onAcad´emicade Ciencias B´asicas,UJAT Km. 1, Carretera Cunduac´an{Jalpade M´endez C.P. 86690, Cunduac´anTabasco, M´exico Carlos Cabrera Instituto de Matem´aticasde la UNAM, Unidad Cuernavaca Av. Universidad s/n. Col. Lomas de Chamilpa C.P. 62210, Cuernavaca, Morelos, M´exico (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(1). The boundary of Kc is called the Julia set Jc. ^ When we consider the polynomial Pc in the extended complex plane C = C [ f1g, 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 B¨ottcher 0 map, φc : Uc ! fz 2 C : jzj > Rg; such that φc(1) = 1; φc(1) = 1 and 2 φc ◦Pc = [φc] . If the critical point 0 is in Kc, then Kc is connected and Uc = CnKc, 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 = fc 2 C : Kc is connectedg: ^ ^ A. Douady and J.H. Hubbard defined the map ΦM : C n M ! C n D, given by ΦM (c) = φc(c) and they proved that ΦM is an analytic isomorphism satisfying 0 ΦM (1) = 1 and ΦM (1) = 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 Bl´e. 6183 6184 GAMALIEL BLE´ AND CARLOS CABRERA For θ 2 T = R=Z, the external ray of argument θ of the Mandelbrot set is the curve −1 2πiθ RM (θ) = ΦM fre : r > 1g : −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 2 W: In fact, in this case Pc is hyperbolic for all c 2 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 2 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 2 N fixed, such that Pc has an attracting cycle of k period k for every c 2 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 2 M in the boundary of a hyperbolic component W has well defined internal and external arguments, [9, 4]. The parameter c 2 @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 2 M \ R; the root of a primitive hyperbolic component. Furthermore, if γ is irrational and satisfies an asymmetrical Diophantine condi- tion 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 hy- perbolic component of the interior of M then the root c 2 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 W : M ! MW ; such that W (0) = c0, W (W0) = W and @MW ⊂ @M: For all c in M the point W (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 2 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 2 @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 ^ 2 @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 ar- guments in the parabolic parametersc ^ 2 @W0 and ct 2 @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 2 M: In particular, if θc ; θc are the external arguments in a parabolic point c 2 W; then + 1.
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