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External Arguments of Douady Cauliflowers in the Mandelbrot Set

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External Arguments of Douady Cauliflowers in the Mandelbrot Set External arguments of Douady cauliflowers in the Mandelbrot set M. Romera*, G. Pastor, G. Álvarez, and F. Montoya Instituto de Física Aplicada, Consejo Superior de Investigaciones Científicas Serrano 144, 28006 Madrid, Spain Near to the cusp of a cardioid of the Mandelbrot set, except for the main cardioid, there is a sequence of baby Mandelbrot sets. Each baby Mandelbrot set is in the center of a Douady cauliflower, a decoration constituted by an infinity of minute Mandelbrot sets and Misiurewicz points linked by filaments. A Douady cauliflower appears to have a complicated structure, and how the external rays land without intersecting in its Misiurewicz points and minute Mandelbrot sets is not really well known. In this work we study the Douady cauliflowers, giving a binary tree model to calculate the binary expansions of the external arguments of both the main cardioid of each minute Mandelbrot set and Misiurewicz points, and creating a horsetail model that explains the arrangement of its corresponding external rays. Keywords: Mandelbrot set, external arguments, external rays, cauliflowers. 1. Introduction In Fig. 1 we can see six details of the Mandelbrot set that we name Mandelbrot medallions, all of them with rotational symmetry. The unquestionable beauty of these Mandelbrot medallions have interested to many researchers as Branner [1] (a), Philip [2] (b), Misiurewicz [3] (c), Munafo [4] (d), Romera and Negrillo [5] (e), and Douady et al. [6] (f). A Mandelbrot medallion presents an infinity of decoration levels around its center. Each decoration level is formed by an infinity of minute Mandelbrot sets and Misiurewicz points with different rotational symmetry. So, the first, second, third... levels have rotational symmetries of orders two, four, eight... In the center of a Mandelbrot medallion there is a baby Mandelbrot set, not a Misiurewicz point, and for this cause the rotational symmetry of a Mandelbrot medallion in not perfect and it must be considered as a quasi-rotational symmetry. * Corresponding author. Tel: +34 915631284; fax: +34 914117651. E-mail address: [email protected] (M. Romera) 2 A Mandelbrot medallion is a very complicated figure. However, there is a type of Mandelbrot medallions first studied by Douady et al. [6], see Fig. 1 (f), which presents less complexity. According to Douady, Buff, Devaney and Sentenac [6] when one looks closely at the Mandelbrot set M at the neighborhood of the cusp point c0 ≠ 14 of a cardioid one can observe a sequence ()M n of small copies of M in M, baby Mandelbrot sets, tending to the cusp point as we shall see graphically in Fig. 2. Fig. 2 (a) shows the Mandelbrot set. Figs. 2 (b) and (c) are enlargements of the squares marked in the previous figures. Let us see the neighborhood of the cusp point of a cardioid in Fig. 2 (c), represented with more detail in the enlargements of Figs. 2 (d) and (e). A sequence of baby Mandelbrot sets tending to this cusp point can clearly be seen in Fig. 2 (e). Each baby Mandelbrot set M n is embedded in the center of a decoration that has been named imploded cauliflower by Douady et al. [6] and we call Douady cauliflower. We can see one of these Douady cauliflowers in Fig. 2 (f), which is an enlargement of the Douady cauliflower inside the square of Fig. 2 (e). The decoration presents an infinity of levels. Each level of the Douady cauliflower is formed by an infinity of minute Mandelbrot sets linked by filaments and located at spirals. The centers of these spirals are Misiurewicz points. The first level, Fig. 2 (g), resembles the 2 1 Julia set of z ++4 ε (for ε > 0 small) and we name it the level one of the Douady cauliflower. The levels two, Fig. 2 (h), three, Fig. 2 (i), four... of the Douady cauliflowers are the same object duplicated, quadruplicated, octuplicated... and wrapped around M n . "Baby Mandelbrot sets are born in cauliflowers" is the title of Ref. [6]. In the early 80's Douady and Hubbard were the authors of the external argument theory in M [7-9]. Douady himself [10] divulged the external argument theory as follows: Imagine a capacitor made of an aluminum bar shaped in such a way that its cross-section is M, placed along the axis of a hollow metallic cylinder. Set the bar at potential 0 and the cylinder at a high potential. This creates an electric field in the region between the cylinder and the bar. An electric potential function is also established in this region. Assume that the radius of the cylinder is large (with respect to the chosen unit), that its height is large compared to its radius and that the length of the bar is equal to this height. We restrict our attention to the plane perpendicular to the axis of the cylinder, through its middle. In this plane, the electric potential defines equipotential lines enclosing the set M (which is the cross section of the bar). Following the electric field, one gets field-lines, called the external rays of M. Each external ray starts at a point x on the boundary of M, and reaches a point y of the 3 great circle which is the cross section of the cylinder (practically at infinity). The position of y is identified by an angle, called the external argument of x with respect to the M set. In Fig. 3 the Mandelbrot set is shown with some of its equipotential lines and external rays. The rational numbers in the figure correspond to the values of the external arguments of the points of M where the external rays land. The unity for external arguments is the whole turn, not the radian. It is often useful to consider the binary expansion (sequence of the binary 1 1 digits) of a rational external argument. So, we have 7 = .001 and 6 = .001. As is known, the binary expansion of a rational external argument with odd denominator is periodic, and the binary expansion of a rational external argument with even denominator is preperiodic. In Fig. 4 we can see six Douady cauliflowers, drawn with different enlargements, corresponding to the sequence of the baby Mandelbrot sets tending to the cusp point of the period-4 cardioid located at −+0.15652…… 1.03224 i (see Fig. 2 (c)). The periods of the baby Mandelbrot sets of Fig. 4 are 7 (the smallest period in the sequence), 11, 15, 63, 119 and 3499 (the largest period that we are able to measure in the sequence). Note that a Douady cauliflower can hardly be seen when the period of the corresponding baby Mandelbrot set is low. A model of the imploded cauliflowers has been described in [6] from a mathematical point of view. In this work we will describe the structure of the Douady cauliflowers from a graphical and experimental point of view, with the aid of a model starting from the observation of a lot of computer experiments. The model is based in the external arguments following Douady and Hubbard [7-10]. According to Schleicher [11], many of the fundamental mathematical results about the Mandelbrot set have been described in the famous "Orsay notes" [8] which have never been published, and it is not always easy to pinpoint a precise reference even within these notes. The Douady cauliflower that surrounds a baby Mandelbrot set shows a lot of minute Mandelbrot sets and also evident Misiurewicz points corresponding to the centers of its eye- catching spirals. In this work we will give a model that calculates the binary expansions of the external arguments of both cusp points of the baby or minute Mandelbrot sets and Misiurewicz points of a given Douady cauliflower. We have verified the model, when it has been possible, using a useful computer program due to Jung [12]. In Sections 2 and 3 we will introduce the composition of two pairs of binary expansions and the harmonics of a pair of binary expansions. These two concepts will be used 4 in Section 4 for the calculation of the binary expansions of baby Mandelbrot sets, minute Mandelbrot sets and Misiurewicz points in Douady cauliflowers. 2. Composition of two pairs of binary expansions According to [13] a key point in Douady and Hubbard's study of M is the theorem that every parabolic point c0 ≠ 14 in M is the landing point for exactly a pair of external rays with external arguments θ1 and θ2 [8]. Graphically, a parabolic point of M is a tangent point or a cusp point. As usual, we write θ1 and θ2 in the form θ11= .a and θ22= .a , where .a1 and .a2 are the binary expansions of the external arguments, and then we write the pair of external arguments of the parabolic point as ().,aa12 . with ..aa12< . ′′ Let ().,aa12 . and ().,aa12 . be the binary expansions of the external arguments of two parabolic points of the Mandelbrot set. We define the compositions C0 and C1 of these pairs of binary expansions by the expressions ′′ ′′ ().,aa12 .〈〉 C 0() ., aa 12 . = ().,aa11 . aa 12 , (1) ′′ ′′ ().,aa12 .〈〉 C 1() ., aa 12 . = ().,aa21 . aa 22 . (2) ′′ The output pair ().,aa11 . aa 12 can define one hyperbolic component or not, i.e., this pair can correspond to either two rays landing at the same parabolic point or two rays landing ′ ′ at two different parabolic points. The same occurs with the pair (.,aa21 . aa 22) . We can verify it computing the kneading sequences of the pair of external arguments [14] and/or drawing the corresponding external rays using a computer program [12]. According to our computer experiments, we have observed that Eqs.
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