# Are Mandelbrot sets useful in any way

## Julia quantities and Mandelbrot quantities

Transcript

1 Julia sets and Mandelbrot set 1. Complex numbers There are quadratic equations that cannot be solved in the set R of real numbers, for example the equation (1) x + 1 = 0 x = 1. Because there is no such thing as a number x R which, multiplied by itself, equals 1. A number with this property must therefore be from a different world than R. It is called i and defines () i = 1. Then the solution of Eq. (1) write (3) x = 1 x = 1 x = 1 x = i x = i. Another example of a quadratic equation that has no solutions in R is (4) xx + 5 = 0. The solution formula (pq-formula), if one considers i = 1, results in x = x = x = 1+ 4 x = 1 4 xx = 1+ = i 1 xx = 1 = 1 i 4 1 Numbers of the form z = x + iy with x, y R are called complex numbers, x is the real part, y is the imaginary part of z. The number i is called an imaginary unit. Mathematically correct, complex numbers are defined, for example, as ordered pairs of numbers (a 1, a) over R, for which equality, (component-wise) addition and multiplication are explained. The multiplication is defined according to (5) (a 1, a) (b 1, b) = ((a 1 b 1 a b), (a 1 b + a b 1)). With these links, the number pairs (a 1, a) form a body, whereby the pairs (a, 0) can be mapped uniquely to the numbers a R. With the notation (a, 0) = a and (0, 1) = i, the complex numbers are obtained in the form a + bi. The set of complex numbers is abbreviated with C. It is astonishing that (1) one does not have to invent any further numbers in order to be able to solve algebraic equations of a higher degree, () calculates with complex numbers in the same way as with real numbers, assuming that i = 1 is observed complex numbers, the real and imaginary parts are processed separately. That is, if z = x + iy and w = u + iv are given, (6) z ± w = (x + iy) ± (u + iv) = (x ± u) + i (y ± v). When multiplying, the distributive law must be observed and i = 1 must also be set: (7) zw = (x + iy) (u + iv) = xu + ixv + iyu + i yv = (xu yv) + i (xv + yu). The possibility of being able to solve algebraic equations of any degree is bought at the price of the fact that complex numbers cannot be arranged. That is, between two 1

2 complex numbers z and w, there is no greater or smaller relation z> w or z

3 (11) and (1) r = x + yy φ = arctan, x if x 0. If x = 0, then φ = π / for y> 0 and φ = π / for y <0. For x = y = 0, φ is not defined .. Iteration and trajectory If one squares the complex number z 0 = 0.8 + 0.4i, one obtains (0.8 + 0.4i) = (0.8 + 0.4i) ( 0.8 + 0.4i) = 0.64 + 0.8 0.4i + 0.16i = 0.64 + 0.64i 0.16 = 0.48 + 0.64i. This number is called z 1. The square of z 1 = 0.48 + 0.64i is 0.6144i and shall be called z. Also z is squared, the result 0.3453 0.0i is called z 3, etc. If one continues this calculation, one obtains a sequence of points zn in the complex number plane, which is called the forward orbit B +. of the point z 0 under the map R (z) = z is called 1. The continuous squaring is described either by the recursion formula (13) z = z n + 1 n or by iterating the map R (z) = z. Starting with z 0, then z 1 = R (z 0), z = R (z 1) = R (R (z 0)), etc. The n-times iteration of R is written R n such that (14) z = nn R z) applies. (0 Note that R (z) denotes the functional equation in the figure, R (in bold) denotes the set of real numbers. In addition, R n, the n-times successive execution of R, should not be to the nth power be confused by R. Let z = R (z) be a map CC and z 0 C, then the sequence z 0, R (z0), R (R (z0)), ... is called the forward orbit ) of the point z 0 under the map R (z). The elements of the sequence are sometimes combined to form the set {z, R (z), R (R ()), ...} + B (z0) = 0 0 z0 The following table (Table 1) shows the first elements of the forward paths under R (z) = z for three different starting points z 0. The paths differ in their behavior for n from z 0 = 0.8 + 0.4i, zn tends towards 0, the starting point z 0 = 0.8 + 0.6i leads to zn = 1 for all n, while z 0 = 0.8 + 0.9i one Orbit results for the zn beyond all limits grows. 3rd

4 Table 1 Forward trajectories of the figure R (z) = z for different starting points z 0 nznnznnzn 0 + 0.8 + 0.4i 0 + 0.8 + 0.6i 0 + 0.8 + 0.9i 1 + 0.48 + 0.64i 1 + 0.8 + 0.96i 1 0.17 + 1.44i 0., 6144i 0,, 5376i, 044 0.489i 3 0.3453 0.0i 3 + 0.419 0.9066i 3 + 3.941 + , 00i 4 + 0,, 151i 4 0.6439 0.7651i, 5 + 15.78i 5 0,. 015i 5 0,, 9853i 5 116,, 7i 6 0,, i 6 0.9416 0.3367i 6 0 ,,,, i 7 + 0,, 6340i 7 + 0,, The starting points z 0, whose paths escape to infinity, are combined to form the set E of escapees. The path R n (z) remains restricted for all other starting points. They form the set P of prisoners. In the present case, E is the outside of the unit circle and P is the inside of this circle including the edge. This definition can be generalized to other mappings z = R (z): Let z = R (z) be a map CC and its iteration zn = R n (z 0) the path of a starting point z 0 C, then {zz for} E = n 0 n is the number of escapees. All other points of the complex plane belong to the set P = C \ E of prisoners. The dichotomy of the complex plane means that there is a boundary between the areas E and P. In the case of the mapping R (z) = z this is a coherent line, namely the unit circle z = 1. All starting points with z 0 = 1 have orbits that neither tend towards zero nor grow beyond all limits, but forever on the unit circle stay. The set of such limit points is called the Julia set after the French mathematician G. Julia (). Only starting points z 0 that lie exactly on this limit have forward orbits that belong to the limit point or Julia set. All other starting points, even those with only a minimal distance from the unit circle, are guided by the mapping R (z) = z in paths that approach the zero point or the point infinitely distant. In that regard, the Julia crowd is a repulsive crowd. The figure shows the forward trajectories of three starting points, the first of which is only slightly inside the unit circle, the second exactly on the unit circle and the third just outside the unit circle. These paths were calculated with the program ForwOrb.java. 4th

5 (a) z 0 = 0,, 598 i (b) z 0 = 0,, 600 i (c) z 0 = 0,, 60 i Fig.Forward trajectories of the figure R (z) = z in the complex number plane for three different starting points z 0. (a) z 0 = 0.800 + 0.598i, little within the unit circle (z 0 = 0.998801), (b) z 0 = 0.800 + 0.600i, i.e. z 0 = 1.000000, exactly on the unit circle, (c) z 0 = 0.800 + 0.60i, just outside the unit circle (z 0 = 1.00101). Forward orbit (a) approaches zero, orbit (b) remains on the unit circle, and orbit (c) disappears in the infinitely distant. The ForwOrb.java program also calculates the forward trajectories for the iteration of the mapping R c (z) = z + c considered below. Here c is a (constant) complex number, i.e. c C. The Julia sets for maps of this kind are not simple structures like the unit circle in the case c = 0. 5

6 3. Fixed point, attractor If one chooses the starting point z 0 = 1 in the mapping R (z) = z, the iteration results in z 1 = 1, z = 1, etc. That means, the forward path B + only contains the starting point z 0. Such a point is called the fixed point of the iteration. Further fixed points of R (z) = z are z 0 = 0 and z 0 =. A point z 0 mapped into itself under R (z), i.e. i.e., for which + R (z 0) = z 0 or B () = {} is called the fixed point of the figure R. z 0 z 0, It has already been mentioned that the points z 0, which in the figure R ( z) = z belong to the set P of prisoners, have forward trajectories that pile up at the point z = 0. On the other hand, the point z = 0 is a fixed point of the iteration. Fixed points can therefore be accumulation values of forward orbits. In this case, they are called attractors (Latin attractor, attract). The iteration of the mapping R (z) = z has the point z = as a further attractor. Because at this point the forward trajectories of the points z 0, which belong to the set E of escapes, accumulate. The set of starting points z 0 whose trajectories lead to one and the same attractor for a given iteration is called the catchment area or basin of the attractor. For R (z) = z, for example, the set P (i.e. the interior of the unit circle) is the catchment area of the attractor z = 0, while the set E (i.e. the area outside the unit circle) is the catchment area of the attractor z = (Fig. 3 ). Fig. 3 For each grid point z 0 of the complex number plane, the forward trajectory was calculated during iteration with R (z) = z. It either ends at the point z0 = 0 + 0i (red) or approaches the infinitely distant point. The trajectories of all starting points z 0, which tend towards z0 = 0 + 0i (white points), form the catchment area or basin of the attractor z0 = 0 + 0i. It is the inside of the unit circle. The orbits of all other starting points (black points) tend towards the infinitely distant point, that is, towards the attractor z =. the point z1 = 1 + 0i (green) is a fixed point of the iteration, that is, R (1 + 0i) = 1 + 0i applies. 6th

7 The third fixed point of R (z) = z, i.e. H. the point z = 1 is not an attractor. Like all other points of the unit circle, it has the property of pushing orbits away from starting points that are in its vicinity (either in the direction z = 0 or in the direction z =). It is therefore called the repulsive fixed point 3. Let Q be a subset of C (i.e. Q C) and z = R (z) be a mapping C C with the fixed point a. If a common accumulation value of the forward trajectories B + (z 0) of all points z 0 Q, then a is called an attractor of the map z = R (z). The set Q is called the catchment area (basin) of the attractor a. 4. Julia set In the case R (z) = z there are two attractors z = 0 and z = whose catchment areas meet at the common boundary line z = 1, the unit circle. If one changes the recursion rule to (15) z = z c n + 1 n + or, in another notation, to R (z) = z c, c n + where c is a (constant) complex number, a new situation arises. A comparatively simple case is c = 1. The table shows the path of the point z 0 = 0.6 + 0i for this c. Table path of the point z 0 = 0.6 + 0i in the iteration R 1 (z) = z 1. nznnznnznnznnzn 1 0.6 5 0,,,,,,,,,,,,,,,,,, 00000 Der Point z 0 = 0.6 + 0i obviously belongs to the set P c of prisoners, because the amount of zn is limited. The sequence does not strive towards a single attractor, but ultimately oscillates back and forth between the two accumulation points z = 0 and z = 1. It leads, as they say, into the attractive cycle γ = {0, 1}. If you start with one of these points, you get the second as a picture and vice versa. This means that for the mapping R 1 (z) = z 1, R 1 (0) = 1 and R 1 (1) = 0. It is an (attractive) cycle with the period. One can show that all starting points belonging to the set P c are attracted to this cycle. Figure 4 shows the catchment area of this attractor or attractive cycle. Its edge is not a smooth geometric curve like the unit circle in the case of R (z) = z, but a line with bubble-like bays and constrictions. Like Fig. 3, the figure was created with the AttrBassin.java program. 7th

8 Fig. 4 Catchment area of the attractor or attractive cycle {0 + 0i, -1 + 0i} of the iteration R 1 (z) = z 1. The edge of the catchment area (the basin) is bulged and constricted like a bubble. Even with the greatest magnification, it does not become "smooth". In general, there are attractors or attractive cycles for each iteration R c (z) = z + c. Their properties depend on the value of the parameter c. One can show that one of the attractors is always z =. That is, for every c C there is the set E c of the breakouts, which is composed of the points z 0, whose orbits go towards the infinitely distant point. The Julia set is the edge of the catchment area of this attactor; i.e., the edge of the set E c. Apart from special cases such as c = 0, this edge is a line that does not become smooth even with the greatest magnification, but rather remains strangely jagged and ramified. It is called a fractal 4th definition (Julia set): Let z 0 C be a starting point in the complex plane whose forward path is z 1 = zn c (with n N 0 and c C) n + + or R c ( z) = z + c is defined, and E c is the amount of breakouts for this parameter c. Then the boundary E of E c is called the Julia set J c for the mapping R c (z) = z + c with the parameter c. c Every point of the complex plane can be an attractor or an element of an attractive cycle. Table 3 shows numerical values for some parameters c. 8th

9 Table 3 Attractors or attractive cycles for a number of parameters c. The attractor z = is not listed. No. c Attractors or attractive cycles 1 0 {0} 0.5 + 0.5i {0,, 751i} 3 0,, 56508i {0.50 + 0.3897i} 4 1 {0; 1} 5 0.1 + 0.74i {0.565i; 0.053i; 0,, 7400i} 5. Scanning method for displaying Julia sets The AttrBassin.java program is primarily used to display the position of the attractors or attractive cycles. The catchment areas of the attractors are not shown very precisely because of the coarse grid. Therefore, the edges of E c, that is, the associated Julia sets, are not easy to see in the display on the screen. In order to show the structures of the edge of E c more precisely, the area of the complex plane in which one suspects this edge is covered with a very dense grid of points z 0. For the iteration with R c (z) = z + c is sufficient it is to consider the section

10 gibein c; // parameter c give in maxiter; // maximum number of iterations give a limit; // upper limit for z * z for all points z with

11 6. Julia sets by inverse iteration A second method of representing Julia sets results from the attempt to compute the ancestors or archetypes of a given point z 0 of the complex plane. The sequence or set of points z from which z 0 can have arisen through repeated application of the mapping R c is called the backward path B (inverse orbit) from z 0 under R c. Let z = R (z) be a mapping CC and z 0 C, then the set k {z R (z) = z for k 0, 1 ,, ...} B (z) = c 0 0 = the backward path (English inverse orbit) of the point z 0 under the map R (z). In practice there is a problem that the inverse mapping to R c is ambiguous. If one writes the map R c: w = z + c, then z z = w c = + w c z = w c. That is, for a given image w of the mapping R c there are two archetypes. They are of the same amount, but differ in sign. Since it is not possible to keep records of all possible ancestors, the capacity of the computer would not be sufficient, one of the archetypes is selected randomly with each step backwards. In this way at least some of the sources of point z 0 are reached. It is astonishing that these sources are located near the edge of E c, i.e. H. pile near points of the Julia set. Indeed, the following applies: The backward trajectory of each point z 0 of the complex plane contains points that come as close as desired to points of the Julia set. Conversely, one can evidently reach any (arbitrarily given) point w of the complex plane from any point z 0 in the vicinity of the Julia set by forward iteration. One can even prove the even stronger statement: If z is an element of the Julia set J c for the map R c, then for every neighborhood U ε (z) of z, however small, there is an iterate R cn such that an arbitrary predetermined point w C is the image under R c of a point z 0 from this environment. That is, there exists an n with the property w = R z0) for z Uε (z) nc (0, roughly speaking, the iterates R cn tend to smear arbitrarily small neighborhoods of the Julia set over the entire complex plane In this sense, J c is a repulsive set. Algorithm for the representation of the Julia set by inverse iteration: For a large number of randomly selected points z 0 of the complex number plane, the backward trajectories are calculated, which are obtained by inverting the mapping R c (z) = z + c. The position of the path point after a sufficient number of (backward) iterations is marked by white coloring the associated pixel

12 n: = 10,000; // number of backwards iterations z: = z 0; // give a randomly chosen initial value in c; repeat n times if random (1) <0.5 // random selection of one of the two then z: = sqrt (z c) // solutions of z = w - c otherwise z: = - sqrt (z c); color point z white // z is finally near J c end repeat. This algorithm is implemented in the JuliaIIM.java program. The Julia set, shown in Figure 6, was calculated using this program. It turns out that the backward trajectories of the randomly selected points z 0 do not end with the same probability at all points of the edge. In the constriction points, for example, no pixels are colored white. Fig. 6 For many randomly selected points z 0 of the complex number plane, the backward trajectories are calculated, which result from the reversal of the mapping R c (z) = z + c - here for c = i. The position of the path point after 100 (backward) iterations is marked by coloring the associated pixel white. All of these orbital points calculated by backwards iteration are in the vicinity of the Julia set. 7. Mandelbrot set Julia sets can have very different forms. For example, for c = 0, J is the unit circle, c = for J gives the interval [; ] the real axis. In addition, there are deformed circles (with a jagged edge), e.g. B. for c = 0,, 56508i, also 1

13 deformed, jagged circles with constrictions (for example for c = 1) and deformed intervals, i.e. H. jagged lines with no interior (c = i). A completely different kind of Julia sets can be discovered if one compares the representations for c = 0.18 + 0.67i and c = 0.11 + 0.67i (Fig. 7). The Julia set, which belongs to the first-mentioned parameter, is a continuous line, while the other falls into a cloud of dust. It is totally disjointed, i. H. a Cantor crowd. (a) (b) Fig. 7 Two Julia sets for different values of the parameter c. (a) Connected Julia set for c = 0.18 + 0.67i. Note that the Julia set is the edge of the black colored area. (b) Disconnected Julia set for c = 0.11 + 0.67i. An incoherent Julia set breaks down into "dust grains" that can hardly be represented by computer graphics. Here the specks of dust are surrounded by white "collars" so that you can guess where they are. Obviously, the value of the parameter c also decides here which of the two cases is present. The question arises for which values of c there are connected Julia sets J c and what the set of these c values looks like in the complex plane. The set of (complex) numbers c, for which J c is connected, is named after B. Mandelbrot, who discovered them and presented them for the first time in computer graphics. 5. The set of all numbers c C for which the corresponding Julia set J c is connected is called the Mandelbrot set M. That is, M {c CJ is connected} =. c The representation of the Mandelbrot set (for example on the computer screen) therefore requires for each c (from the area of interest of the complex c-plane) to check whether the associated Julia set J c is connected or a Cantor dust cloud . The decision is made on the basis of a criterion provided by the following theorem by Julia and Fatou (P. Fatou, French mathematician). 13th

14 Fundamental theorem of the Mandelbrot set: The Julia set J c for the parameter c is connected if and only if the forward path B c + (0) of the point z 0 = 0 is bounded. Then c belongs to the Mandelbrot set M: {0, c, c + c, (c + c) + c,} is bounded + c M Bc (0) = computer graphic representation of the Mandelbrot set In practice, the decision is made whether B c + (0) is bounded or not, as in the case of the sampling method in the representation of the Julia sets. Again, a maximum number of iterations (maxiter) and an upper limit (limit) are given for the square z of the path point. If z is smaller than the bound after the given maximum number of iterations, one assumes that the value of c just examined belongs to the Mandelbrot set M. Otherwise it is calculated as the complement C \ M. In this case, the number n of iterations is a measure of the reciprocal value of the escape velocity with which the orbit B c + (0) tends towards infinity. The pixel corresponding to c is colored black, similar to the case of the Julia set, if c is an element of the Mandelbrot set, otherwise the color is chosen according to the number n of iterations until the limit for z is reached. Algorithm for generating the Mandelbrot set for R c (z) = z + c according to the sampling method: For all values c in the range 5

15 Figure 8 shows the amount of Mandelbrot that the program displays on the screen. Note that the edge of M does not become smoother even when enlarged. This property is characteristic of a fractal. The branching of the edge is emphasized by the fact that areas with the same escape speed in the area outside M are colored according to n (n = number of iterations until the limit for z is reached). An example of this is shown in Fig. 9, others can be found in the literature 6, 7. Fig. 8 Mandelbrot set in the complex number level (black colored area). The Mandelbrot set is the set of all numbers c for which the Julia set J c is connected. The edge of this set is a fractal, that is, an unbroken line that does not become "smooth" even at any high magnification. Fig. 9 Section from the edge of the Mandelbrot set The section shows the area of the complex number plane in the interval Re (c) and Im (c)

16 Notes and literature The designation Vorwärtsbahn and all other technical terms are taken from the book Heinz-OttoPeitgen, Hartmut Juergens and Dietmar Saupe: Fractals for the Classroom, Springer-Verlag, New York, 199. A list of all computer programs used here can be found in Appendix witer below. However, R (1) = (1) = 1. Iterations that start from other starting points do not end up at z = 1. The attraction basin of the fixed point z = 1 is therefore only a single point on the complex plane, namely z = 1 The books mentioned under 1) and 5) offer a detailed and complete classification of the fixed points of a July set. A comprehensive introduction to the field of fractals is the book Heinz-Otto Peitgen, Hartmut Jürgens and Dietmar Saupe: Fractals for the Classroom, Springer-Verlag, New York, 199. It consists of two volumes, volume 1 with the Subtitle Introduction to Fractals and Chaos, volume with Complex Systems and Mandelbrot Set. Very well suited for self-study (school level), but with computer programs in the BASIC language, which is no longer common today. One of the first books on fractals is Benoit B. Mandelbrot: Fractals Form, Chance, and Dimension, Freeman, San Francisco, mathematically more demanding than the book Fractals for the Classroom. The work in which Mandelbrot first published a graph of his set is Mandelbrot, Benoit B .: Fractal aspects of the iteration of z λz (1 z) for complex λ and z, Annals New York Academy of Sciences 357 (1980 ), A look back at the history of the discovery is Mandelbrot's article Fractals and the Rebirth of Iteration Theory in the book Peitgen, Heinz-Otto and Peter H. Richter: The Beauty of Fractals Images of Complex Dynamical Systems, Springer-Verlag , Heidelberg, 1986, pages In his book The Fractal Geometry of Nature, Freeman, San Francisco, 1977, Mandelbrot presents his quantity on page 188. Peitgen, Heinz-Otto and Peter H. Richter: The Beauty of Fractals Images of Complex Dynamical Systems, Springer-Verlag, Berlin Heidelberg New York Toronto, This book contains many colored illustrations of Juliamenets, of the Mandelbrotset and of excerpts from them, not only those that belong to the iteration with R (z) = z + c. The mathematics of July sets is also dealt with in detail. In addition, there is a contribution by Mandelbrot on the historical development of the theory of fractals (see above) and an article by A. Douady on Julia sets and the Mandelbrot set. Peitgen, Heinz-Otto and Dietmar Saupe (Editors): The Science of Fractal Images, Springer-Verlag, New York Berlin Heidelberg London Paris Tokyo, contributions by various authors, among others. also about the creation of fractal landscapes. (Imparatively formulated) algorithms are offered for almost all problems. They can be transferred into common programming languages with little effort. 16

17 Appendix List of the computer programs used 1 ForwOrb.java Calculates the forward path of a point z 0 of the complex plane by iterating the mapping R (z) = z + c. Input: z 0 = (x 0, y 0), output: path point after each iteration step. The path is also shown in the Gaussian numerical plane. Examples: Fig. AttrBassin.java Determines the catchment area (basin) of the attractors or the attractive cycles of the mapping R (z) = z + c. Input: cx, cy. The pool is displayed on the screen. Examples: Fig. 3 and Fig JuliaBSM.java Determines the Julia set of the mapping R (z) = z + c according to the scanning method (algorithm see text) and displays it on the screen. Input: cx, cy. Examples: Fig. 5 and Fig JuliaIIM.java Determines the Julia set of the mapping R (z) = z + c using the inverse iteration method (algorithm see text) and displays it on the screen. Input: cx, cy. Example: Abb MandelbrotBSM.java Displays the Mandelbrot set of the figure R (z) = z + c on the screen. Works according to the sampling method (algorithm see text). Input: boundaries of the complex plane area to be displayed. Examples: Fig. 8 and Fig

18 JuliaMandelbrot.doc 18

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