1

OPERATING WITH EXTERNAL ARGUMENTS

IN THE ANTENNA*

G. Pastor†, M. Romera, G. Alvarez, and F. Montoya

Instituto de Física Aplicada. Consejo Superior de Investigaciones Científicas, Serrano 144, 28006 Madrid, Spain

Abstract: The external argument theory of Douady and Hubbard allows us to know both the potential and the field-lines in the exterior of the Mandelbrot set. Nonetheless, there are no explicit formulae to operate with external arguments, and the external argument theory is difficult to apply. In this paper we introduce some tools in order to obtain formulae to operate with external arguments in the Mandelbrot set antenna. Thus, we introduce the harmonic tool to calculate both the external arguments of the period doubling cascade hyperbolic components and the external arguments of the last appearance hyperbolic components. Likewise, we introduce composition rules applied to external arguments that, with the aid of the concept of heredity, allows the calculation of all the external arguments that constitutes the family tree of a given external argument.

PACS: 05.45.Ac, 47.52+j

1. Introduction As is known, Douady and Hubbard studied the Mandelbrot set M [1,2], a bounded set of complex numbers, by computing both the potential created by the set and the external arguments of the points on its boundary [3-5]. Douady himself divulged the external argument theory as follows [6]. Imagine a capacitor made of an

* This paper was published in Physica D, 171 (2002), 52-71. † Corresponding author. Tel. (34) 915 61 88 06, Fax: (34) 914 11 76 51, Email: [email protected]

2 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 great circle which is the cross section of the cylinder (practically at infinity). The position of y is identified by an , called the external argument of x with respect to the M set. If there are several accesses to x from outside of M (for instance, if M is tree-like and x is a branch point of M), then there is one external ray in each access and the point x has several external arguments. In Fig. 1 the M 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 , not the radian. An external ray starting at a point x on the boundary of M is identified by the external argument of x, and all the points of this external ray have the same external argument. As is well known, the Mandelbrot set is defined as the set of points c of the complex plain for which the sequence cc, 222+++ c, () c c c ,… does not tend to ∞ [7]. This sequence is originated by the iteration of the complex quadratic polynomial

2 Pzc ()=+ z c. Let us consider the above model of Douady and Hubbard. The potential Uc() of a point c in the interior of the capacitor but outside the Mandelbrot set, is given by the Green’s function

∞ 1 c Uc()=+log c log 1 + , (1) ∑ 2n n−1 2 n=1 Pcc ()

0 12 222 where Pcc ()= c, Pcc ()=+ c c, Pcccc = ()++… [8]. The base of the logarithm does not matter. In fact, eq.(1) ignores the factor of proportionality that depends on the 3 potential of the hollow metal cylinder. We use logarithms to the base 2, according to other authors [9]. Eq.(1) gives potential zero at points of the boundary of the Mandelbrot set. For a large c, Eq. (1) reduces to Uc( ) = log c. The locus of points in the having all of them the same potential is a simple closed curve surrounding the Mandelbrot set. This curve is an equipotential line that consists of values of c for which 0 escapes to ∞ at a fixed rate [7]. As is known, the external argument θ (c) of the external ray that pass through a point c is given by the argument of the function Φ(c) [10]

2− n ∞ ⎡⎤c Φ=()cc⎢⎥1 + . ∏ n−1 2 n=1 ⎣⎦⎢⎥Pcc ()

Taking into account that arg(ab )=+ arg( a ) arg( b ) we have

∞ 1 ⎡⎤c θ ()cc=+arg () arg⎢⎥ 1 + , (2) ∑ 2n n−1 2 n=1 ⎣⎦⎢⎥Pcc () where arg ()c denotes the principal value of the argument of a , which takes values between −π and π . In this paper we draw external rays using an algorithm from Jung [11,12]. The locus of points in the complex plane that have all a same external argument is an external ray. Douady and Hubbard found an algorithm to compute the two external arguments of a point c on the Mandelbrot set antenna [5], as we explain next. We take a point c* next to c, over the antenna, obtaining the sequence of arguments {arg(cccccc*****2* ), arg(22+++ ), arg(( ) ),…}. Next we replace each argument by “0” (if it is greater than 0 and less than π ) or “1” (if it is greater than π and less than 2π ). In this way we obtain the binary expansion of one of the external arguments of c, that is easy to convert at a rational number (its external argument). The another external argument of c is obtained in the same manner, taking a new point c* next to c, under the antenna. Thus, the external arguments of the c = −1.543689012… ⎛⎞57 are 512 and 712, which we write as the pair ⎜⎟, . ⎝⎠12 12 4

In this work we will develop formulae to operate with external arguments in the Mandelbrot set antenna. With these formulae we can better understand the raison d’être, the physical meaning, of a given pair of external arguments corresponding to a point on the Mandelbrot set antenna. For example, as we will see later, the external arguments of the above Misiurewicz point can be obtained, without knowing its parameter value c, starting from the well known external arguments of both the root point of period-3 ⎛⎞34 ⎛⎞12 midget ⎜⎟, and the root point of period-2 disk ⎜⎟, . We will obtain the external ⎝⎠77 ⎝⎠33 arguments of this Misiurewicz point as the rightward composition of the external arguments of the only period-3 antenna midget with the external arguments of the period-2 disk an infinite number of times.

Each one of the Mandelbrot set discs has a root point which connects with a bigger disc or with a cardioid. Each one of the Mandelbrot set cardioids, except for the main cardioid, has a cusp point from which a filament that connects it with the set emerges. Both root points and cusp points are notable points of the boundary of the Mandelbrot set, in the same way as Misiurewicz points which usually are either ramification nodes or tips of the filaments. Other notable points, as the Myrberg-Feigenbaum point, are not studied here.

External arguments are not measured in radians, but as fractions of complete turns. Using this unit, most of the notable points of the Mandelbrot set boundary have rational external arguments. Fig. 2 shows the neighborhood of the Mandelbrot set antenna with four examples of notable points and their external rays. The first one on the right is a root point (where the main cardioid and the first disc of the period-doubling cascade join together) which has two rational external arguments with odd denominator, ⎛⎞12 ⎜⎟, . Even though external arguments come from the root point, we also use this pair ⎝⎠33 of external arguments in order to denominate the whole disc that is born at this root point. The second one is the Myrberg-Feigenbaum point, with two irrational external arguments that we shall not study here. The third one is a Misiurewicz point which has ⎛⎞57 two rational external arguments with even denominator, ⎜⎟, . The fourth and last ⎝⎠12 12 one is a cusp point (where the only antenna period-3 cardioid is born) which has two 5

⎛⎞34 rational external arguments with odd denominator, ⎜⎟, . Again, even though external ⎝⎠77 arguments come from the cusp point, we also use this pair of external arguments in order to denominate the hole cardioid that is born at this cusp point.

As we can easily verify, a rational number with an odd denominator is periodic under the function doubling modulo 1. For example, by doubling 1/5 we obtain 1/5, 2/5, 4/5, 3/5, 1/5, … which is periodic. Likewise, a rational number with even denominator is preperiodic under doubling modulo 1. For example, by doubling 1/6 we obtain 1/6, 2/6, 4/6, 2/6, … which is preperiodic. As is known, in the Mandelbrot set we have periodic and preperiodic points. Root and cusp points are periodic, but Misiurewicz points are preperiodic. Furthermore, the action of this function is to double the argument (for big values of z). This is why periodic points have periodic external arguments (rational ones with odd denominator), and preperiodic points have preperiodic external arguments (rational ones with even denominator). The Mandelbrot set is without any doubt the more representative nonlinear dynamical system. These nonlinear dynamical systems have been broadly treated in all disciplines, especially in physics because of their wide range of applications in a large number of nonlinear problems. Thus, the first physically motivated study of chaotic dynamics was presented in 1963 by Lorenz [13], and in 1970 Ruelle and Takens suggested that turbulent flow might be an example of dynamical chaos [14]. In following years dynamical systems were applied to almost any branch of the physics from pattern formation in natural systems [15] to fluid mechanics [16]. Nowadays, the interest on this topic continues as we can see for example in [17]. Therefore, we reckon that studying nonlinear dynamical systems in the widest possible manner should positively contribute to enrich physical research. We have studied one-dimensional (1D) quadratic maps [18-22] by means of the hyperbolic components [7], Misiurewicz points [4,18,23-25] and all the other notable points of the Mandelbrot set which intersect with the real axis. Fig. 2 shows the neighborhood of this Mandelbrot set subset, that we call “antenna”. This subset has an enormous interest since it is equivalent to a 1D quadratic map, and can be used to study these last maps. In previous papers we have studied hyperbolic components and Misiurewicz points of the antenna by using their symbolic sequences [26-28] (or keading sequences according to other authors [29]). In this paper we pretend to study 6 the antenna by using the external arguments of A. Douady and J. H. Hubbard instead of the symbolic sequences we used to use. In a previous paper, we based on the MSS harmonics [30] to introduce other harmonics [20], what we called there Fourier harmonics (because they were very similar) though later we have called them F-harmonics or simply harmonics. By using this tool we could calculate all the structural components (and the Misiurewicz points that separates them) of the 1D quadratic maps; i. e., we could determine the structure of such maps [20]. In the first part of this paper we pretend to apply the harmonic tool for the case of external arguments, in such a manner that we can calculate the external arguments of the structural hyperbolic components and the Misiurewicz points that separate them. I. e., we want to calculate the structure of the antenna that intersects with the real axis, but now given in external arguments. Likewise, in other previous paper [21] we introduced the concept of heredity in 1D quadratic maps. Indeed, as we showed there, given a hyperbolic component we can calculate the ancestral path that joins all its ancestors, and from these ancestors we can calculate all the descendants of the hyperbolic component, i.e. the family tree. In order to do so, we need to use the composition rules of the symbolic sequences that were introduced by us. In this paper we also introduce new composition rules, now applied to external arguments. With these composition rules for external arguments we will be able, by using the concept of heredity cited before, of calculating all the external arguments that constitutes the family tree of a given external argument. To begin with, let us see how to operate with external arguments of the Mandelbrot set antenna in a similar way as we did with symbolic sequences in our previous papers. In such a way we can calculate the external arguments of the hyperbolic components and Misiurewicz points of the antenna.

2. MSS harmonics of a hyperbolic component. Let us consider a period p hyperbolic component of the antenna. If it is a disc, it is connected to the antenna through its tangent point (root point); and, if it is a cardioid, through its cusp. When we start from a period p hyperbolic component, and we progress through its period doubling cascade, as is well known we find discs whose periods are 21 p , 22 p , 23 p , … . Metropolis, Stein and Stein (MSS) introduce the concept of 7 harmonic in one-dimensional unimodal maps. By extension, we call “MSS harmonics” of a hyperbolic component of the Mandelbrot set (here of the antenna) to the set constituted by itself and the discs of its period doubling cascade. The MSS harmonics were an important achievement because, starting from the symbolic sequence (or pattern) of a period p hyperbolic component we can calculate the pattern of all the discs of its period doubling cascade. Thus, the ith MSS harmonic is the ith disc of the period doubling cascade, which has a period 2i p , and therefore a pattern that is 2i times long the original. As we said before, we have applied these concepts to the case of external

⎛⎞aa12 arguments. Indeed, let ⎜⎟, be the normalized external arguments of a ⎝⎠2121pp−− period p hyperbolic component. The normalized external arguments of the MSS harmonic of order i of this hyperbolic component can be calculated by means of the following expression:

⎛⎞aa ⎛⎞aa1(ii ) 2( ) H ()i 12,,= , i ≥1, (1) MSS ⎜⎟⎜⎟ ⎝⎠bb⎝⎠ b()ii b () where:

2i−1 p 2i−1 p 2i p aa1()ii=+ 1(1)−−2 a 2(1) i , aa2(ii )=+ 2(− 1)2 a 1( i− 1) , b()i = 21− and

p aa1(0)= 1 , aa2(0)= 2 , b(0) =−21.

Example 1

The external arguments of the Mandelbrot set main cardioid are 0 ,1 . The first ( 11) four MSS harmonics of 0 ,1 are the external arguments of the first four discs of the ( 11) period doubling cascade of such a main cardioid:

(1) ⎛⎞⎛⎞01 12 H MSS ⎜⎟⎜⎟,,= , ⎝⎠⎝⎠11 33

(2) ⎛⎞⎛01 6 9 ⎞⎛⎞ 23 H MSS ⎜⎟⎜,,== ⎟⎜⎟ ,, ⎝⎠⎝1 1 15 15 ⎠⎝⎠ 5 5 8

(3) ⎛⎞⎛0 1 105 150 ⎞⎛ 7 10 ⎞ H MSS ⎜⎟⎜,,== ⎟⎜ , ⎟ and ⎝⎠⎝1 1 255 255 ⎠⎝ 17 17 ⎠

(4) ⎛⎞⎛0 1 27030 38505 ⎞⎛ 106 151 ⎞ H MSS ⎜⎟⎜,,== ⎟⎜ , ⎟. ⎝⎠⎝1 1 65535 65535 ⎠⎝ 257 257 ⎠

In Fig. 3(a) we have separatedly depicted each one of the external arguments of the

()∞ ⎛⎞01 first four discs of the period doubling cascade of the main cardioid. H MSS ⎜⎟, are the ⎝⎠11 external arguments of the Myrberg-Feigenbaum point. This notable point has not rational external arguments. As is known, the series 1 3= 0.01, 2 5= 0.0110 ,

7 17= 0.01101001, 106 257= 0.0110100110010110 , … tends to the Thue-Morse constant θMF = 0.4124…, which is the smaller of the external arguments of the Myrberg-Feigenbaum point. Obviously, the series 23, 35, 10 17 , 151 257 … tends to

1−θMF .

Example 2

The external arguments of the only period 3 cardioid of the antenna, located at

c =−1.754877666..., are 3 , 4 . The first four MSS harmonics of 3 , 4 are the ( 77) ( 77) external arguments of the first four discs of the period doubling cascade of such a period 3 cardioid:

(1) ⎛⎞⎛⎞⎛⎞3 4 28 35 4 5 H MSS ⎜⎟⎜⎟⎜⎟,,,==, ⎝⎠⎝⎠⎝⎠7 7 63 63 9 9

(2) ⎛⎞⎛3 4 1827 2268 ⎞⎛⎞ 29 36 H MSS ⎜⎟⎜,,== ⎟⎜⎟ ,, ⎝⎠⎝7 7 4095 4095 ⎠⎝⎠ 65 65

(3) ⎛⎞⎛3 4 7485660 9291555 ⎞⎛ 1828 2269 ⎞ H MSS ⎜⎟⎜,,== ⎟⎜ , ⎟ and ⎝⎠⎝7 7 16777215 16777215 ⎠⎝ 4097 4097 ⎠

(4) ⎛⎞⎛3 4 125588544014115 155886432696540 ⎞⎛ 7485661 9291556 ⎞ H MSS ⎜⎟⎜,,== ⎟⎜ , ⎟. ⎝⎠⎝7 7 281474976710656 281474976710656 ⎠⎝ 16777217 16777217 ⎠

In Fig. 3(b) we have separately depicted each one of the two external arguments of the first three discs of the period doubling cascade of the period 3 cardioid. Likewise, 9

()∞ ⎛⎞34 H MSS ⎜⎟, , whose external arguments are irrational again, and corresponds to the ⎝⎠77 equivalent Myrberg-Feigenbaum point, is the limit of the former external arguments.

()i The external arguments of the harmonic H MSS of a period p hyperbolic component placed on the antenna, calculated by means of eq. (1), are in what we denominate normalized form. The normalized form of a pair of external arguments of a hyperbolic component is characterized by a denominator 2p − 1, where p is its period. However, as we have seen in the two former examples, the normalized form can be simplified by dividing the numerator and denominator of each one of them by 212i−1 p − , obtaining what we denominate the irreducible form.

()i ⎛⎞aa12 ⎛⎞⎛⎞aa1(ii ) 2( )αα 1( i ) 2( i ) Let us suppose that H MSS ⎜⎟,,==⎜⎟⎜⎟ ,, where ⎝⎠bb⎝⎠⎝⎠ bii b β iβ i

⎛⎞aa1(ii ) 2( ) ⎛⎞αα1(ii ) 2( ) ⎜⎟, is the normalized form and ⎜⎟, is the irreducible form. In this ⎝⎠bbii ⎝⎠βiiβ case, we have the following property for the discs of the period doubling cascade (which are calculated by means of the MSS harmonics) of the antenna cardioids:

⎛⎞⎛⎞αα1(ii++ 1) 2( 1)aa 1( i )++11 2( i ) ⎜⎟⎜⎟,,= . ⎝⎠⎝⎠ββii++11 bb i++22 i

()i ⎛⎞aa12 i. e., by knowing the normalized form of H MSS ⎜⎟, , we can directly give the ⎝⎠bb

(1)i+ ⎛⎞aa12 irreducible form of H MSS ⎜⎟, and vice versa, as can clearly be seen in the two ⎝⎠bb former examples.

3. Fourier harmonics of a hyperbolic component As we have already pointed out before in [20], we have introduced the Fourier harmonics, or F-harmonics, applied to the pattern of a hyperbolic component of a 1D quadratic map. These F-harmonics have a different meaning that the MSS harmonics. Indeed, as we saw before, the order i MSS harmonic corresponds to the ith disc of the period doubling cascade which has a period 2i p , and therefore a pattern with a length

2i times the original. However, the Fourier harmonics, except for the first harmonic, are all of them last appearance cardioids in the corresponding chaotic band [20], and the 10 order i F-harmonic has a period (ip+ 1) , and therefore has a pattern with a length i +1 times the original. As we did in the case of the MSS harmonics, we shall introduce now the F- harmonics of the external arguments of an antenna hyperbolic component. Again we shall see that these harmonics, except for the first one, correspond to the external arguments of the last appearance cardioids of the corresponding chaotic bands.

3.1. External arguments of the Fourier harmonics of a hyperbolic component.

⎛⎞aa12 Let ⎜⎟, be the normalized external arguments of a period p hyperbolic ⎝⎠2121pp−− component. By definition, the normalized external argument of the order i F-harmonic of this hyperbolic component is given by:

ji=−11 ji =− ⎛⎞ip jp ip jp ⎜⎟aa122222++∑∑ aa 21 ()i ⎛⎞aa12⎜⎟jj==00 H F ⎜⎟,,= ip++11 ip . (2) ⎝⎠2121pp−−⎜⎟21()−− 21() ⎜⎟ ⎝⎠

It is easy to verify that the external arguments of both the first F-harmonic and the first MSS harmonic of a hyperbolic component are the same. I. e.:

(1)⎛⎞aa12 (1) ⎛⎞ aa 12 HHFMSS⎜⎟,,= ⎜⎟. ⎝⎠bb ⎝⎠ bb

Example 1

The external arguments of the Mandelbrot set main cardioid are 0 ,1 . The four ( 11)

0 1 first F-harmonics of ()11, are external arguments of last appearance cardioids, except for the first one which is the first disc of its period doubling cascade. These values are:

(1) ⎛⎞⎛⎞01 12 H F ⎜⎟⎜⎟,,= ⎝⎠⎝⎠11 33

(2) ⎛⎞⎛⎞01 34 H F ⎜⎟⎜⎟,,= ⎝⎠⎝⎠11 77

(3) ⎛⎞⎛01 7 8 ⎞ H F ⎜⎟⎜,,= ⎟ ⎝⎠⎝1 1 15 15 ⎠ 11

(4) ⎛⎞⎛0 1 15 16 ⎞ H F ⎜⎟⎜,,= ⎟ ⎝⎠⎝1 1 31 31 ⎠

In Fig. 4(a) we show the four first former values. As we can see, indeed the first harmonic corresponds to the first disc of the period doubling cascade, while the following three correspond to the period 3, 4 and 5 last appearance cardioids.

Example 2

The external arguments of the only period 3 antenna cardioid, located at

c =−1.754877666..., are 3 , 4 . The four first F-harmonics are the external arguments ( 77) of the first disc of the period doubling cascade (with period 6) and the three last appearance cardioids (with periods 9, 12 and 15) of such a period 3 cardioid:

(1) ⎛⎞⎛⎞3 4 28 35 H F ⎜⎟⎜⎟,,= ⎝⎠⎝⎠7 7 63 63

(2) ⎛⎞⎛3 4 228 283 ⎞ H F ⎜⎟⎜,,= ⎟ ⎝⎠⎝7 7 511 511 ⎠

(3) ⎛⎞⎛3 4 1823 2267 ⎞ H F ⎜⎟⎜,,= ⎟ ⎝⎠⎝7 7 4095 4095 ⎠

(4) ⎛⎞⎛3 4 14628 18139 ⎞ H F ⎜⎟⎜,,= ⎟ ⎝⎠⎝7 7 32767 32767 ⎠

In Fig. 4(b) we show the three first former values. As we can see, indeed the first harmonic corresponds to the first disc of the period doubling cascade, while the following two correspond to the period 33⋅ and 34⋅ last appearance cardioids.

aa By definition, 21− is the normalized increment of the argument of a hyperbolic bb

⎛⎞aa12 component whose normalized external arguments are ⎜⎟, . By symmetry, the ⎝⎠bb addition of the two external arguments of an antenna hyperbolic component is equal to the unity. As we can see, the addition of the external arguments of each one of the main 12 cardioid F-harmonics is equal to the unity, since such F-harmonics are placed on the antenna. Besides each one of the main cardioid F-harmonics is, among all the hyperbolic components with the same period, that which has the smaller normalized increment of the argument. Thus, on the Mandelbrot set antenna there are three period 5 hyperbolic components whose normalized external arguments are 13, 18 , 14 ,17 ( 31 31) ( 31 31) and 15, 16 ; however, only the last one, with the smaller normalized increment of the ( 31 31) argument, is a F-harmonic. This pair of external arguments correspond indeed to the fourth main cardioid F-harmonic.

3.2. Limit of the Fourier harmonics When we studied F-harmonics by means of the symbolic sequences [20], we saw that the order ∞ F-harmonic of a cardioid is the tip of such a cardioid, while the order ∞ F-harmonic of a disc of the period doubling cascade is a merging point. We shall see next that for the case of external arguments the same happens. The external argument of the order ∞ F-harmonic of a hyperbolic component

⎛⎞aa12 whose external arguments are ⎜⎟, , aa21> , can be calculated as: ⎝⎠2121pp− −

pp ⎛⎞aa ⎛⎞aaaa1212()21−+ +( 21 −) H ()∞ 12,,= ⎜⎟ (3) F ⎜⎟pp ⎜⎟pp pp ⎝⎠2121−−⎝⎠22()−− 1 22() 1

Eq. (3) has been calculated starting from eq. (2) when i tends to the infinity.

Example 1

Let us calculate the order ∞ F-harmonic of the main cardioid, whose external arguments are 0 ,1 . By applying eq. (3): ( 11)

()∞ ⎛⎞⎛⎞01 11 H F ⎜⎟⎜⎟,,= ⎝⎠⎝⎠11 22 that indeed is the only value (double) of the tip located at c = −2 , as we can see in Fig. 4(a).

Example 2 13

Let us calculate the order ∞ F-harmonic of the only period 3 cardioid of the antenna, located at c =−1.754877666..., whose external arguments are 3 , 4 . By ( 77) applying eq. (3):

()∞ ⎛⎞⎛⎞3 4 25 31 H F ⎜⎟⎜⎟,,= ⎝⎠⎝⎠7 7 56 56 that indeed are the external arguments of the tip of the cardioid 3 , 4 , located at ( 77) c =−1.790327491..., as we can see in Fig. 4(b). In these two examples we have seen that the order ∞ F-harmonic of a cardioid is the tip of such a cardioid. In the following section we shall see several examples to show that the order ∞ F-harmonic of a disc of the period doubling cascade is a merging point.

3.3. Application to the harmonic structure of the antenna. F-harmonics are a very powerful tool in 1D quadratic maps since starting from the main cardioid, C, we can generate the successive chaotic bands and calculate the pattern of any of their structural components (which are the last appearance hyperbolic components) [20]. Let us see this statement more carefully, though we advise to turn to [20] to see it in depth. Let us note that we use to denominate the whole hyperbolic component by means of the pattern of its superstable orbit in the same way as we denominate such a hyperbolic component by means of the external arguments of its root point or its cusp point. As we can see in [20], we started from the period-1 main cardioid placed in the periodic region or Feigenbaum region. The first F-harmonic of the main cardioid is the period-2 disc of the period-doubling cascade. All the other F-harmonics of the main

0 cardioid are cardioids placed in the period-1 ( 2 ) chaotic band B0 and are the last appearance cardioids of this chaotic band. If we consider the period- 20 main cardioid as a “gene” G0 , then the F-harmonics of the gene G0 generate the structure of the period-

0 2 chaotic band B0 . The limit of this band B0 is the Misiurewicz point with preperiod

2 and period 1, mM02,1= , that is the antenna tip placed in c = −2 . 14

We are in the same scenario in the case of external arguments. Indeed, if we start from the external argument of the main cardioid and we calculate the successive F- harmonics according to eq.(2), we have:

(1) ⎛⎞⎛⎞01 12 H F ⎜⎟⎜⎟,,= ⎝⎠⎝⎠11 33

(2) ⎛⎞⎛⎞01 34 H F ⎜⎟⎜⎟,,= ⎝⎠⎝⎠11 77

(3) ⎛⎞⎛01 7 8 ⎞ H F ⎜⎟⎜,,= ⎟ ⎝⎠⎝1 1 15 15 ⎠

(4) ⎛⎞⎛0 1 15 16 ⎞ H F ⎜⎟⎜,,= ⎟ ⎝⎠⎝1 1 31 31 ⎠ As we show in Fig. 5, the first F-harmonic corresponds to the period- 21 first disc of the period-doubling cascade. All the other F-harmonics of the main cardioid are

0 cardioids placed in the period-1 ( 2 ) chaotic band B0 and are the last appearance cardioids of this band. Again, we can consider the period- 20 main cardioid as a “gene” ⎛⎞01 G0 = ⎜⎟, , then the F-harmonics of the gene G0 generate the structure of the period- ⎝⎠11

0 ()∞ 2 chaotic band B0 . The limit of this band B0 is H F (G0 ) calculated according to eq.(3):

()∞ ⎛⎞⎛⎞01 11 H F ⎜⎟⎜⎟,,= ⎝⎠⎝⎠11 22 that indeed corresponds to the Misiurewicz point with preperiod 2 and period 1,

mM02,1= , that is the antenna tip placed at c = −2 . At the tip both external arguments 11 coincide in only one, but double, in such a way that + =1 (see Fig.6). 22

⎛⎞01 1 The first F-harmonic of G0 = ⎜⎟, is not a cardioid but a period- 2 disc placed ⎝⎠11 in the Feigenbaum region, that we shall call “bridge pattern”. Let us remember what happens when this disc is used as a new gene in [20]. We can consider this “bridge pattern”, which is the first disc of the period-doubling cascade, as a new gene G1 . Then,

1 we have that the F-harmonics of the gene G1 generate the structure of the period- 2 chaotic band B1 (and a new bridge pattern that will be the second disc of the period- 15

doubling cascade, the new gene G2 ). The limit of this band B1 is the Misiurewicz point

0 mM13,1= , placed at c =−1.543689012 , that separates the period- 2 chaotic band B0

1 and the period- 2 chaotic band B1 . For external arguments, if we start from the external argument of the gene ⎛⎞12 G1 = ⎜⎟, and we calculate the successive F-harmonics according to eq.(2), we have: ⎝⎠33

(1) ⎛⎞⎛⎞⎛⎞12 6 9 23 H F ⎜⎟⎜⎟⎜⎟,,,== ⎝⎠⎝⎠⎝⎠3 3 15 15 5 5

(2) ⎛⎞⎛1 2 26 37 ⎞ H F ⎜⎟⎜,,= ⎟ ⎝⎠⎝3 3 63 63 ⎠

(3) ⎛⎞⎛1 2 106 149 ⎞ H F ⎜⎟⎜,,= ⎟ ⎝⎠⎝3 3 255 255 ⎠ As we show in Fig. 5, the first F-harmonic corresponds to the period- 22 second ⎛⎞12 disc of the period-doubling cascade. All the other F-harmonics of the gene G1 = ⎜⎟, ⎝⎠33

1 are cardioids placed in the period- 2 chaotic band B1 and are the last appearance ⎛⎞12 cardioids of this band. Hence, the F-harmonics of the gene G1 = ⎜⎟, generate the ⎝⎠33

1 ()∞ structure of the period- 2 chaotic band B1 . The limit of this band B1 is H F ()G1 calculated according to eq.(3):

()∞ ⎛⎞⎛⎞12 5 7 H F ⎜⎟⎜⎟,,= ⎝⎠⎝⎠3 3 12 12 that indeed corresponds to the Misiurewicz point with preperiod 3 and period 1, ⎛⎞57 mM13,1==⎜⎟, , placed at c = −1.543689012 , that separates the period-1 chaotic ⎝⎠12 12 band B0 and the period-2 chaotic band B1 (see Fig. 6). If we consider the new “bridge pattern”, which is the second disc of the period- ⎛⎞⎛⎞69 23 doubling cascade, as a gene G2 ==⎜⎟⎜⎟,, (see Fig. 5), we have that the F- ⎝⎠⎝⎠15 15 5 5 harmonics of the gene G2 generate the external arguments of the structure of the

2 period- 2 chaotic band B2 (and another bridge pattern that will be the third disc of the 16

⎛⎞⎛⎞105 150 7 10 period-doubling cascade, the new gene G3 ==⎜⎟⎜⎟,,). The limit of this ⎝⎠⎝⎠255 255 17 17

(1) ⎛⎞⎛⎞99 141 33 47 band B2 corresponds to the Misiurewicz point mM25,2==⎜⎟⎜⎟,, = , ⎝⎠⎝⎠240 240 80 80 that separates the period-2 chaotic band B1 and the period-4 chaotic band B2 (see Fig. 6). Let us see that in Fig. 5 we always depict the irreducible form of the arguments, although we have to use the normalized form to operate. Generalizing, by applying eq. (2), the F-harmonics of the disc n of the period-

(1) doubling cascade, the gene GGnFn= H ()−1 , generates the external arguments of the

n last appearance cardioids of the period- 2 chaotic band Bn , and a bridge disc of the

(1) ()∞ period-doubling cascade, the gene GGnFn+1 = H (). Likewise, H Fn()G is a Misiurewicz point m = M [19], a primary separator (or band-merging point) of n 21,2ii+ −1 the chaotic bands Bn−1 and Bn , where external arguments are calculated by eq. (3). This double procedure (discs and chaotic bands generation) continues indefinitely, and both meet at the Myrberg-Feigenbaum point [31], periodic discs on the right and chaotic bands on the left. Thanks to this procedure the whole harmonic structure of a 1D quadratic map was rigorously calculated in [20] by using symbolic sequences and now by using external arguments. In the same way as we have calculated the harmonic structure of the main cardioid, we can calculate the harmonic structure of anyone of the antenna midgets (in Fig. 5 of [20] we can see a sketch of the harmonic structure of the only period 3 midget of the antenna, whose cusp is placed at c = −1.754877666...) . Likewise, we can calculate the harmonic structure of any rightward 1D quadratic map, as for example the logistic map (see Fig. 6 of [20]). Therefore, the method is general for any type of 1D quadratic maps.

4. Composition of the external arguments. By using the tools we have introduced up to now, we can calculate the external arguments of the structural components (those of smaller increment of the argument or of last appearance) and the Misiurewicz points being tips or being merging points. That is what we denominated in [20] the structure of a 1D quadratic map, that here is the structure of the antenna, and that more or less is its framework or skeleton. But for each one of these hyperbolic components and Misiurewicz points there are an infinity more 17 that have not been treated so far. In [20] we introduced other new tools (composition rules), and in another of our papers [21] we introduced new concepts (heredity) that make possible to calculate the descendants of any hyperbolic component if we know all the ancestors of such a hyperbolic component. Let us see if the above considerations can be applied to the antenna external arguments.

4.1. Leftward composition and rightward composition.

In [20] we give the composition rules of the symbolic sequences of two hyperbolic components: the leftward and the rightward composition rules, that mnemonically we denominate “lero rule” and “relo rule”. Let us see here the equivalent rules for the case of external arguments.

⎛⎞aa12 Let ⎜⎟, be the normalized external arguments of a hyperbolic ⎝⎠2121pp−−

⎛⎞aa12′ ′ component of period p (augend), and let ⎜⎟′′, be the normalized external ⎝⎠2121pp− − arguments of a hyperbolic component of period p’ (addend). By definition, the normalized external arguments of the leftward composition are given by:

pp′′ ⎛⎞⎛⎞aa12 aa 1′′ 2⎛⎞22 aaaa 1221++ ′ ′ ⎜⎟⎜⎟,,+=′′⎜⎟ ′ , ′ (4) ⎝⎠⎝⎠2121p−− p 21212 p −− p⎝⎠() pp++ − 12 () pp − 1

Likewise, by definition, the normalized external arguments of the rightward composition are given by:

pp′′ ⎛⎞⎛⎞aa12 aa 1′′ 2⎛⎞22 aaaa 1122++ ′ ′ ⎜⎟⎜⎟,,+=′′⎜⎟ ′ , ′ (5) ⎝⎠⎝⎠2121p−− p 21212 p −− p⎝⎠() pp++ − 12 () pp − 1

Example

Let the augend be the pair of normalized external arguments 13, 18 that ( 31 31) correspond to the period 5 hyperbolic component α3 of the Fig. 7(a) and (b), and let the addend be the period 2 first disc of the period doubling cascade whose external arguments are 12, . The normalized external arguments of the leftward composition ( 33) 18

⎛⎞⎛⎞⎛⎞13 18 1 2 54 73 are obtained by applying eq. (4): ⎜⎟⎜⎟⎜⎟,,+= , that, indeed, are the ⎝⎠⎝⎠⎝⎠31 31 3 3 127 127 normalized external arguments which correspond to the period 7 hyperbolic component 7a of the Fig. 7(a) and (b). Likewise, the normalized external arguments of the rightward composition are obtained by applying eq. (5): ⎛⎞⎛⎞⎛⎞13 18 1 2 53 74 ⎜⎟⎜⎟⎜⎟,,+= ,that, indeed, are the normalized external arguments ⎝⎠⎝⎠⎝⎠31 31 3 3 127 127 which correspond to the period 7 hyperbolic component 7b of the Fig. 7(a) and (b).

4.2. External arguments of a Misiurewicz point

⎛⎞aa12 Let ⎜⎟, be the normalized external arguments of a hyperbolic component ⎝⎠2121pp−−

⎛⎞aa12′′ and ⎜⎟′′, the normalized external arguments of a heredity transmitter ⎝⎠2121pp−− ancestor [21]. The external arguments of the leftward composition of the two hyperbolic

⎛⎞⎛⎞aa12 a 1′ a 2′ components are:⎜⎟⎜⎟,,+ ′′, that are calculated according to eq. ⎝⎠⎝⎠21212pp−− p − 121 p − (4). Likewise by applying eq. (4), the external arguments of the leftward composition of the former result with the heredity transmitter ancestor are: ⎡⎤⎛⎞⎛⎞⎛⎞aa aa′′ aa ′′ 12,,++ 12 12 ,. If we repeat the process, we ⎢⎥⎜⎟⎜⎟⎜⎟pp pp′′ pp ′′ ⎣⎦⎝⎠⎝⎠⎝⎠2121−− 2121 −− 2121 −−

⎡⎤⎡⎤⎛⎞⎛⎞⎛⎞⎛⎞aa aa′′ aa ′′ aa ′′ have: 12,,,,+++ 12 12 12. ⎢⎥⎢⎥⎜⎟⎜⎟⎜⎟⎜⎟pp pp′′ pp ′′ pp ′′ ⎣⎦⎣⎦⎝⎠⎝⎠⎝⎠⎝⎠2121−− 2121 −− 2121 −− 2121 −− In the limit, these infinite leftward compositions give the external arguments of a Misiurewicz point located on the left of the hyperbolic component. If the heredity transmitter ancestor is the first harmonic of the gene of the chaotic band where the hyperbolic component is placed, the Misiurewicz point in a characteristic one (it has the same period as the chaotic band) [18]. Otherwise, is a non-characteristic one [18]. As it is easy to prove mathematically, the former infinite repetition of the leftward composition gives:

19

⎛⎞aa′′ aa++21 ⎡⎤⎡⎤⎛⎞⎛⎞⎛⎞aa a′′ a a ′′ a ⎜⎟12pp′′ 12,,+++= 12 12 ,… 21−− , 21 (6) ⎢⎥⎢⎥⎜⎟⎜⎟⎜⎟pp pp′′ pp ′′ ⎜⎟ p p ⎣⎦⎣⎦⎝⎠⎝⎠⎝⎠2121−− 2121 − − 2121 − − ⎜⎟ 2 2 ⎝⎠ In the same way, infinite rightward compositions give the external arguments of a Misiurewicz point located on the right of the hyperbolic component. As it is also easy to prove mathematically, the infinite repetition of the rightward composition gives: ⎛⎞aa′′ aa++12 ⎡⎤⎡⎤⎛⎞⎛⎞⎛⎞aa a′′ a a ′′ a ⎜⎟12pp′′ 12,,+++= 12 12 ,… 21−− , 21 (7) ⎢⎥⎢⎥⎜⎟⎜⎟⎜⎟pp pp′′ pp ′′ ⎜⎟ p p ⎣⎦⎣⎦⎝⎠⎝⎠⎝⎠2121−− 2121 − − 2121 − − ⎜⎟ 2 2 ⎝⎠

Example 1 The limit of the rightward composition of the external arguments of the period 3 ⎛⎞34 ⎛⎞12 hyperbolic component α2 , ⎜⎟, , with its heredity transmitter ancestor ⎜⎟, is: ⎝⎠77 ⎝⎠33

⎡⎤⎡⎤⎡⎤⎛⎞⎛⎞⎛⎞⎛⎞34 12 12 12 ⎛ 5 7 ⎞ ⎢⎥⎢⎥⎢⎥⎜⎟⎜⎟⎜⎟⎜⎟,,++++= , ,… ⎜ , ⎟, which in Fig. 7(a) and (b) is ⎣⎦⎢⎥⎣⎦⎣⎦⎝⎠⎝⎠⎝⎠⎝⎠77 33 33 33 ⎝ 1212 ⎠ the Misiurewicz point M3,1 [24].

Example 2 The limit of the leftward composition of the external arguments of the period 5 ⎛⎞13 18 ⎛⎞12 hyperbolic component α3 , ⎜⎟, , with its heredity transmitter ancestor ⎜⎟, is: ⎝⎠31 31 ⎝⎠33

⎡⎤⎡⎤⎡⎤⎛13 18 ⎞⎛⎞⎛⎞⎛⎞ 1 2 1 2 1 2 ⎛ 41 55 ⎞ ⎢⎥⎢⎥⎢⎥⎜,,,, ⎟⎜⎟⎜⎟⎜⎟++++=… ⎜ , ⎟, which in Fig. 7(a) and (b) ⎣⎦⎢⎥⎣⎦⎣⎦⎝31 31 ⎠⎝⎠⎝⎠⎝⎠ 3 3 3 3 3 3 ⎝ 96 96 ⎠ is the Misiurewicz point M6,1 [24].

4.3. Autocomposition. Harmonics and antiharmonics

Let us consider the case of the leftward autocomposition of the external arguments of a period p hyperbolic component. As we can see, in such a manner we obtain the external arguments of the first F-harmonic of the hyperbolic component. Indeed: 20

pp ⎛⎞⎛⎞aa12 aa 12⎛⎞ aaaa 122122++ ⎜⎟⎜⎟,,+=⎜⎟ , , (8) ⎝⎠⎝⎠21212121pp−− pp −−⎝⎠ 222 p − 1 2 p − 1

pp (1) ⎛⎞aa12⎛⎞ aaaa 122122++ but also: H F ⎜⎟,,= ⎜⎟. ⎝⎠2121pp−−⎝⎠ 222 p − 1 2 p − 1

If we apply the autocomposition again and again we obtain the successive F- harmonics. I. e., the order i F-harmonic of the normalized external arguments

⎛⎞aa12 ⎜⎟, , that are calculated by means of eq. (2), is also the order i ⎝⎠2121pp−−

⎛⎞aa12 autocomposition of ⎜⎟, and, therefore, can be calculated as follows: ⎝⎠2121pp−−

()i aa12⎛⎞⎛⎞⎛⎞ aa 12 aa 12 aa 12 aa 12 H ,,,,,=+++ (9) F pp⎜⎟⎜⎟⎜⎟ pp pp pp pp ()2121−−⎝⎠⎝⎠⎝⎠() 2121 −−( 2121 −−) ( 2121 −−) () 2121 −−

where the addition is repeated i times. Let us consider now the case of the rightward autocomposition of the external arguments of a period p hyperbolic component. As we can see, with this autocomposition we obtain first a period 2p hyperbolic component, the antiharmonic of the period p hyperbolic component, but, after an elemental simplification, we obtain again the period p hyperbolic component. Indeed:

pp ⎛⎞⎛⎞aa aa⎛⎞aa12(21++) ( 21) ⎛⎞ aa 12,,+= 12⎜⎟ , = 12 ,. (10) ⎜⎟⎜⎟pp pp⎜⎟22 p p ⎜⎟ pp ⎝⎠⎝⎠21212121−− −−⎝⎠ 2 − 1 2 − 1 ⎝⎠ 2121 −−

Antiharmonics have already been introduced by Metropolis Stein and Stein in 1976 [30]. Normally, it is thought that antiharmonics have no real existence. However, we reckon that this idea must be revised because, although we have the same value when we simplify, if we do not simplify we have a body, the antiharmonic, with a period which is double than the original one. Therefore, even though placed on the same location, it is different. Besides, it is important to study them because, as we saw in [21], they can be heredity transmitters.

Example 21

Let us calculate the harmonic and the antiharmonic of period 5 hyperbolic ⎛⎞13 18 component α3 (see Fig. 7(b)), whose external arguments are ⎜⎟, (see Fig. 7(a)). ⎝⎠31 31 ⎛⎞⎛⎞⎛13 18 13 18 434 589 ⎞ In order to do so, we apply eqs. (8) and (10): ⎜⎟⎜⎟⎜,,+= , ⎟ is the ⎝⎠⎝⎠⎝31 31 31 31 1023 1023 ⎠ harmonic (10H in Fig. 7(a) and (b)), and ⎛⎞⎛⎞⎛13 18 13 18 429 594 ⎞⎛⎞ 13 18 ⎜⎟⎜⎟⎜,,,,+= ⎟⎜⎟ = is the antiharmonic (α4 =10A in Fig. ⎝⎠⎝⎠⎝31 31 31 31 1023 1023 ⎠⎝⎠ 31 31 7(b)) given in the normalized and irreducible forms. As expected, its normalized form does not correspond to any period 10 existing antenna hyperbolic component. However, its irreducible form coincides with the normalized form of the period 5 original hyperbolic component α3 . All the antiharmonics have this property, though in same cases when we simplify we obtain the original hyperbolic component before reaching the irreducible form.

4.4. Ancestral decomposition of the external arguments.

As we have seen in [21], the pattern of a hyperbolic component of a 1D quadratic map can be expressed as the addition of the gene of the chaotic band where the pattern is, and the heredity transmitters related to the ancestral path that joins all its ancestors. We can see that if we manage external arguments of the antenna hyperbolic components, the same happens. As we know, the normalized external arguments of a period p hyperbolic

⎛⎞aa12 component have the form ⎜⎟, . However, an antenna hyperbolic component ⎝⎠2121pp−− has not arbitrary values a1 and a2 . The possible values are very scarce if we impose the aa condition 12+=1, which points out that the hyperbolic component is on the 2121pp−− antenna. Nevertheless, still there are some values of a1 and a2 that, even if they verify this last condition, do not correspond to any antenna hyperbolic component. When this

⎛⎞aa12 happens, we say that ⎜⎟, are non-legal external arguments. Therefore, we ⎝⎠2121pp−− 22

⎛⎞aa12 can say that legal external arguments are a pair of external arguments ⎜⎟, ⎝⎠2121pp−− where a1 and a2 correspond to existing values of antenna hyperbolic components. A pair of legal external arguments can always be decomposed in the successive additions (toward the left or toward the right) of the structural components of the chaotic band where they are placed. Next, by analyzing this decomposition, we can easily see the ancestral path and the heredity transmitters of the legal external aa arguments. We shall use as legality test (in addition to 12+ =1) the following 2121pp−− conditions: first, the decomposition has to be possible, and second: the decomposition has to correspond to a valid ancestral path. When these conditions are verified, the external arguments are legal ones. If, on the contrary, the decomposition is not possible; or if the decomposition being possible it does not correspond to a valid ancestral path, then the external arguments are non-legal ones. Let us think in the inverse process: let us suppose that we know the external arguments of the i structural components obtained from the decomposition of a pair of legal external arguments. Obviously, if we now compose them (by using the additions in the proper direction obtained from the decomposition) we obtain again the pair of legal external arguments of the beginning. If the structural components have periods p1 ,

p2 , ..., pi , then we know the values of their external arguments, and the decomposition is:

aa aa aa aa ⎛⎛⎛1(1) 2(1) ⎞⎛ 1(2) 2(2) ⎞⎞⎛ 1(3) 2(3) ⎞⎞⎛ 1(ii ) 2( ) ⎞⎛⎞aa12 ,,++ , += ,,(11) ⎜⎜⎜pp ⎟⎜ pp ⎟⎟⎜pp ⎟⎟⎜ pp ⎟⎜⎟pp ⎝⎝⎝2121212111−− ⎠⎝ 2 −− 2 ⎠⎠⎝ 212133 −− ⎠⎠⎝ 2121ii −− ⎠⎝⎠ 2121 −−

where p =+++pp12 pi , and where + is either + or + . Let us see again the decomposition of a pair of external arguments. This decomposition has to verify eq. (11). If what we know is the result, we have to test all the possible cases, discard all that do not lead to the solution, and take the only one that lead us to the correct result, i. e., we test all the possible p = pp12+++ pi . For each one of these decompositions we calculate the possible values of the eq.(11) with all the additions towards the left and towards the right. This is very long and tedious if we do it manually, but it is elementary if we make it by using a computer program. 23

Once this decomposition is obtained, by taking into account the heredity rules that we saw in [21], it is basic to find the ancestral path that joins all the ancestors and determine the heredity transmitters. Now, if we compose the legal external arguments with their heredity transmitters we obtain all the “children” or first generation descendants. By applying the same procedure to each one of its children we obtain all the grandchildren. If we continue uninterruptedly this procedure we obtain the family tree.

4.5. Application to the family tree of external arguments

Let us calculate the descendants of the period 12 hyperbolic component 12h whose ⎛⎞⎛⎞1717 2378 1717 2378 external arguments are P,==⎜⎟⎜⎟12 12 ,, which is a simplified case ⎝⎠⎝⎠2−− 1 2 1 4095 4095 of the example of the section IV of [21]. Fig. 7(a) shows a zone of the antenna with 12h and its ancestors and descendants. Fig. 7(b) shows a sketch of Fig. (a).

⎛⎞⎛⎞1717 2378 1717 2378 Starting from the value of P,==⎜⎟⎜⎟12 12 ,, we calculate the ⎝⎠⎝⎠2−− 1 2 1 4095 4095 decomposition in structural components as we explained in the previous section. The obtained result, given in its simplified form, is:

1232322=++++ (12) where, instead of the well known structural components external arguments, we only indicate the periods and the addition directions. We shall see, with the help of Fig. 7(b) as we describe the process, how with this datum we can find the ancestral path, the ancestors and the heredity transmitters. If the heredity transmitters are known, we can ⎛⎞⎛⎞1717 2378 1717 2378 calculate the descendants of P,==⎜⎟⎜⎟12 12 ,. ⎝⎠⎝⎠2−− 1 2 1 4095 4095 ⎛⎞34 The decomposition (12) starts from the period 3 structural component α2 = ⎜⎟, , ⎝⎠77 of the chaotic band B0 , though this is not the first component of the ancestral path.

⎛⎞34 ⎛⎞01 Before α2 = ⎜⎟, , we have the main cardioid, α0 = ⎜⎟, , and the first disc of the ⎝⎠77 ⎝⎠11 24

⎛⎞12 period doubling cascade, α1 = ⎜⎟, , that are not in B0 , and therefore are not in the ⎝⎠33 ⎛⎞34 decomposition. To the period 3 hyperbolic component α2 = ⎜⎟, , we add rightwards ⎝⎠77 ⎛⎞12 its period 2 ancestor, α1 = ⎜⎟, , and we obtain the next hyperbolic component of the ⎝⎠33 ⎛⎞⎛⎞⎛⎞3 4 1 2 13 18 ancestral path: α3 =+=⎜⎟⎜⎟⎜⎟,, ,, that has period 5 as can be seen in Fig. ⎝⎠⎝⎠⎝⎠7 7 3 3 31 31 7(a) and (b). If we analyze decomposition (12), it seems that now we should compose rightwards this period 5 hyperbolic component and the period 3 structural component.

However, as we know from [21], even though the period 3 hyperbolic component α2 is an ancestor of the period 5 α3 , α2 is not a heredity transmitter, and therefore this composition is not valid. But we can compose rightwards this period 5 hyperbolic component and ++ 3 2 , i. e., we can compose rightwards this period 5 hyperbolic component and itself to give the antiharmonic. Hence we have: ⎛⎞⎛⎞⎛13 18 13 18 429 594 ⎞⎛⎞ 13 18 α4 =+=⎜⎟⎜⎟⎜,, , ⎟⎜⎟ = ,, that is also named 10A in Fig. ⎝⎠⎝⎠⎝31 31 31 31 1023 1023 ⎠⎝⎠ 31 31 7(a) and (b). Let us note that, since it is an antiharmonic, it has two normalized forms: the first one (period 10) really corresponds to the antiharmonic, and the second one (period 5) is the starting hyperbolic component. As we can see in decomposition (12), we end next with the composition +2 , that is a valid one, and we obtain the starting period 12 hyperbolic component: ⎛⎞⎛⎞⎛⎞429 594 1 2 1717 2378 P,=+=⎜⎟⎜⎟⎜⎟ , ,, as we can see in Fig. 7(b) named 12h. ⎝⎠⎝⎠⎝⎠1023 1023 3 3 4095 4095 Now, we want to calculate the first generation descendants (children) of P. In order to do so, as we know from [21], we have to compose, both rightwards and leftwards, P and all the heredity transmitters. Hence, we have to determine the heredity transmitters. In order to do so, let us continue on the ancestral path that joints all the ancestors with

P. This ancestral path is: α12,,,αα 4, as we can see in Fig. 7(b). The period 2 ancestor ⎛⎞12 ⎛⎞34 α1 = ⎜⎟, is a heredity transmitter. The period 3 ancestor α2 = ⎜⎟, is not a ⎝⎠33 ⎝⎠77 25

⎛⎞13 18 heredity transmitter. The period 5 ancestor α3 = ⎜⎟, is neither a heredity ⎝⎠31 31 ⎛⎞⎛⎞⎛⎞13 18 1 2 53 74 transmitter, but its period 7 child ch()α3 =+=⎜⎟⎜⎟⎜⎟ , , , is a heredity ⎝⎠⎝⎠⎝⎠31 31 3 3 127 127 ⎛⎞429 594 transmitter. The period 10 ancestor α4 = ⎜⎟, is neither a heredity transmitter. ⎝⎠1023 1023 ⎛⎞12 We have determined all the heredity transmitters of P (here two, α1 = ⎜⎟, and ⎝⎠33 ⎛⎞53 74 ⎛⎞1717 2378 ch()α3 = ⎜⎟ , ). We compose rightwards and leftwards P,= ⎜⎟ and ⎝⎠127 127 ⎝⎠4095 4095 these two heredity transmitters in order to find the children of P. Thus, we obtain:

⎛⎞⎛⎞⎛⎞1717 2378 1 2 6869 9514 ch1r =+=⎜⎟⎜⎟⎜⎟,, , ⎝⎠⎝⎠⎝⎠4095 4095 3 3 16383 16383 ⎛⎞⎛⎞⎛⎞1717 2378 1 2 6870 9513 ch1l =+=⎜⎟⎜⎟⎜⎟,, , ⎝⎠⎝⎠⎝⎠4095 4095 3 3 16383 16383 ⎛⎞⎛⎞⎛⎞1717 2378 53 74 219829 304458 ch2r =+=⎜⎟⎜⎟⎜⎟,, , ⎝⎠⎝⎠⎝⎠4095 4095 127 127 524287 524287 ⎛⎞⎛⎞⎛⎞1717 2378 53 74 219850 304437 ch2l =+=⎜⎟⎜⎟⎜⎟,, ,. ⎝⎠⎝⎠⎝⎠4095 4095 127 127 524287 524287 If we repeat now the proccess for each one of the children, we obtain the grandchildren, great-granchildren, and so on up to the infinity to obtain all the descendants or the family tree of P.

5. Conclusions

In this paper we have introduced several tools in order to calculate the external arguments of Douady and Hubbard in the Mandelbrot set antenna. We have introduced the MSS harmonics of external arguments of a period-p hyperbolic component. The order i MSS harmonic calculates the external arguments of the ith disc of the period doubling cascade, whose period is 2i p , of such a period-p hyperbolic component. We have introduced the Fourier harmonics (or F-harmonics) of external arguments of a period-p hyperbolic component. The order i F-harmonic calculates the external 26 arguments of the last appearance cardioids in the corresponding chaotic band (except for the first harmonic that calculates the first disc of the period doubling cascade). The order i F-harmonic of a period-p hyperbolic component has a period (ip+ 1) . Besides, we have studied the order ∞ harmonics of both, MSS harmonics and Fourier harmonics. The order ∞ MSS harmonic of the external arguments of a period-p hyperbolic component is a Myrberg-Feigenbaum point, whose external arguments are irrational, that we do not consider here. The order ∞ Fourier harmonic of the external arguments of a period-p hyperbolic component is a Misiurewicz point. The order ∞ F- harmonic of a cardioid is a tip, and the order ∞ F-harmonic disc is a merging-point.

We have introduced the leftward composition and the rightward composition of the external arguments of a period p hyperbolic component (augend) and a period p’ hyperbolic component (addend). With these composition rules for external arguments, and by taking into account the concept of heredity, we can calculate all the external arguments that constitute the family tree of a pair of given external arguments. 27

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