6. The Mandelbrot Set

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1 1 The Mandelbrot Set King of mathematical monsters Im Re Christoph Traxler Fractals-Mandelbrot 1 Christoph Traxler Fractals-Mandelbrot 2 6.1

2 Christoph Traxler Fractals-Mandelbrot 3 Christoph Traxler Fractals-Mandelbrot 4 6.2

3 Christoph Traxler Fractals-Mandelbrot 5 Ordering the Julia Sets J c For each c of f c = z 2 +c there exists a particular Julia set J c The set of all Julia sets is uncountable infinite How can J c be classified with respect to c? Christoph Traxler Fractals-Mandelbrot 6 6.3

4 Ordering the Julia Sets Jc Mandelbrot s idea (1979): Create a map of the complex plane, which shows information about the Julia sets for each complex number c What is the classification criterion? Christoph Traxler Fractals-Mandelbrot 7 Ordering the Julia Sets J c Dichotomy of Julia sets: Julia sets can be seperated into two categories: J c is connected J c is totally disconnected (Cantor dust) Christoph Traxler Fractals-Mandelbrot 8 6.4

5 Ordering the Julia Sets J c Visualize the area of points for which the corresponding Julia set is connected Def.: The set M = {c C J c is a connected set } is called Mandelbrot set Fact: J c is connected if and only if the critical orbit 0 c c 2 +c... is bounded Christoph Traxler Fractals-Mandelbrot 9 Critical Point of J c Def.: If f c (z) = 0, then z is called critical point f c (z) = 2z z = 0 is the critical point for any J c The orbit of the critical point z=0 is called critical orbit If and only if the critical orbit escapes to then J c is Cantor dust Christoph Traxler Fractals-Mandelbrot

6 Calculating the Map Calculate the fate of the critical point z = 0 Iterate z z 2 +c, z 0 = 0 The orbit is not bounded and escapes to if z > 2 J c is totally disconnected c M Approximation of M with the pixel game, accuracy depends on the number of iterations Visualization of the equipotentials of A( ) Christoph Traxler Fractals-Mandelbrot 11 Calculating the Map The pixel game algorithm for M: for(i=0; i<height; i++) for(j=0; j<width; j++) { c = point4pixel(i,j); for(n=0,z=0; n<=nmax; n++) { if(rad(z)> 2) 2) break; z = z*z + c; c; } if(n > NMAX) setpixel(i,j,black); else setpixel(i,j,getcol(n)); } Christoph Traxler Fractals-Mandelbrot

7 Difference between J c and M Julia set f c (z) = z 2 + c c is a constant z is variable examine the orbit z f c (z) f c2 (z)... for each point z Mandelbrot set f(z) = z 2 + c c is variable z 0 = 0 is constant examine the orbit 0 f(0) f 2 (0)... for each c Christoph Traxler Fractals-Mandelbrot 13 Zoom Sequence Christoph Traxler Fractals-Mandelbrot

8 Zoom Sequence Christoph Traxler Fractals-Mandelbrot 15 Zoom Sequence Christoph Traxler Fractals-Mandelbrot

9 Zoom Sequence Christoph Traxler Fractals-Mandelbrot 17 Calculating the Map Faster Divide & conquer approach Quadtree subdivision of the complex plane Calculate the points on the edges of the grid and on the diagonals of each rectangle If all points of a rectangle and of its diagonal have the same value n, then no further subdivision is necessary Subdivision is also terminated if the rectangles are smaller then a pixel Christoph Traxler Fractals-Mandelbrot

10 Calculating the Map Faster Christoph Traxler Fractals-Mandelbrot 19 Encirclement of J c Approximation of J c with deformed discs Start with a disc S r that surrounds J c and calculate the sequence: S k = { z f c (z) S k-1 },with S 0 = S r This sequence of nested deformed discs converges to J c The deformed discs correspond to equipotential curves of A( ) Christoph Traxler Fractals-Mandelbrot

11 Encirclement of J c One of the following conditions is true: A)The whole sequence S n, n, consits of Jordan curves (disc like shapes) B) S 0,...,S k are Jordan curves but not the rest of the sequence S k+1,... Christoph Traxler Fractals-Mandelbrot 21 Encirclement of J c If S k is a Jordan curve, then there are 3 possible cases for S k+1 : 1) S k+1 is also a Jordan curve and the critical point 0 and point c are in the interior of S k+1, condition A is true 0 S k+1 Christoph Traxler Fractals-Mandelbrot

12 S k+1 0 S k+1 0 Encirclement of J c 2) S k+1 is made up of B Jordan curves, which touch exactly at the critical point, condition B is true 3) S k+1 is made up of two disjoint Jordan curves and does not contain the critical point, condition B is true Christoph Traxler Fractals-Mandelbrot 23 Encirclement of J c If condition A is true, then K c contains the critical point and c (their orbits don t escape to ) J c is connected If condition B is true, then K c neither contains the critical point nor c (their orbits escape to ) J c is disconnected Christoph Traxler Fractals-Mandelbrot

13 Encirclement of J c S r S r c S k c S k K c K c Condition A is true Condition B is true Christoph Traxler Fractals-Mandelbrot 25 Aquivalent Definitions of M The Mandelbrot set M can be defined in various ways: M = { c C J c is connected } = { c C c K c } = { c C critical point 0 K c } = { c C critical point 0 A( ) } = { c C f n (c) 2 n = 0,1,2,... } Christoph Traxler Fractals-Mandelbrot

14 Level Sets The encirclement of J c is a decomposition of A( ) into subsets of points which converge to with the same speed These subsets are called level sets Each area of attraction of a periodic point can be decomposed in this way Level sets visualize the behaviour of points under iteration in an area of attraction Christoph Traxler Fractals-Mandelbrot 27 Level Sets Decomposition of an attraction area A(p) of an attractive periodic point p into a level set: Choose a target set T which contains p The level of each point z C is defined as: l c (z) = k, if f ci (z) T, i<k, and f ck (z) T l c (z) = 0, else Level set L k = { z l c (z) = k }, k is the convergence speed of z Christoph Traxler Fractals-Mandelbrot

15 Level Sets Examples: Julia set, Mandelbrot set, p =, T = { z z 1/ε }, ε very small Christoph Traxler Fractals-Mandelbrot 29 Level Sets Target set T arbitrary, for example: {c,z 2 }, A(0) is the unit circle, decomposition of A(0) with T = K c, c M Christoph Traxler Fractals-Mandelbrot

16 Level Sets Level sets as height fields Christoph Traxler Fractals-Mandelbrot 31 Level Sets Level sets as height fields Christoph Traxler Fractals-Mandelbrot

17 Level Sets Level sets as height fields Christoph Traxler Fractals-Mandelbrot 33 Binary Decomposition Extension of the level set method Subdivision of the target set into 2 parts induces a subdivision of the level L k into 2 k+1 parts Iteration of f c (z), z = r(cosϕ + isinϕ) doubles the angle ϕ a loop of a closed curve in L k corresponds to 2 k+1 loops in T The borders of the subdivided parts are good approximations of field lines Christoph Traxler Fractals-Mandelbrot

18 Binary Decomposition For each point z L k : Draw a black point if 0 angle(f ck (z)) π Draw a white point else T ϕ L1 ϕ T L 1 Christoph Traxler Fractals-Mandelbrot 35 Binary Decomposition Christoph Traxler Fractals-Mandelbrot

19 Binary Decomposition Christoph Traxler Fractals-Mandelbrot 37 Properties of M M is connected M contains infinite many copies of itself These copies are connected with the main body by strings The fractal dimension of M is 2 M is a table of content for all connected Julia sets J c Christoph Traxler Fractals-Mandelbrot

20 Properties of M Christoph Traxler Fractals-Mandelbrot 39 Components of M Cardioid - heart shaped region Buds - circle like shapes connected to the cardioid or other buds Two components are connected by only one point, called seed point Copies of M Buds correspond to Julia sets that bound the area of attraction of a periodic point with a specific period Christoph Traxler Fractals-Mandelbrot

21 4 Components of M buds 2 seed points largest copy of M 3 5 cardioid Numbers indicate the period of the periodic point of the corr. J c Christoph Traxler Fractals-Mandelbrot 41 Classes of J c with Respect to M If c is a point of the cardioid then J c is a Jordan curve with a fixpoint (period = 1) P Christoph Traxler Fractals-Mandelbrot

22 Classes of J c with Respect to M If c is a seed point then P is an indifferent periodic point P Christoph Traxler Fractals-Mandelbrot 43 Classes of J c with Respect to M If c lies on the border of M and is no seed point then P is an indifferent periodic point in the Siegel disc Siegel disc P Points of K c rotate on the Siegel disc arround P Christoph Traxler Fractals-Mandelbrot

23 Classes of J c with Respect to M If c lies on a connecting string then is the only fixpoint of J c, K c = J c This class of Julia sets is called dendrites Christoph Traxler Fractals-Mandelbrot 45 Classes of J c with Respect to M If c lies in a copy of M then J c is a combination of the J c1 of the connecting string and the J c2 of the main body where c 2 has the same relative position in the main body Christoph Traxler Fractals-Mandelbrot

24 Classes of J c with Respect to M If c lies outside of M then J c is totally disconnected (Cantor dust), the density of the points decreases with increasing distance to M Christoph Traxler Fractals-Mandelbrot 47 Similarity Relation M has a similar structure in the environment of c as the corresponding J c when magnified at the same position Christoph Traxler Fractals-Mandelbrot

25 Structural Stability of M Example: {C, r c } r c = ((z 2 +c-1) / (2z+c-2 )) 2 r c describes magnetic phase transitions Attractive fixpoints: 1, Examine the orbit 0 r c (c) r 2 c (c)... for all c and draw a map What does M there? Christoph Traxler Fractals-Mandelbrot 49 Structural Stability of M Answer: r c has a local behaviour like f c =z 2 +c M is universal, - it can be discovered in several dynamic systems Given {C, f p }, f arbitrary, p C the map of the parameter space contains a (maybe deformed) copy of M with high probability Christoph Traxler Fractals-Mandelbrot

26 Structural Stability of M Christoph Traxler Fractals-Mandelbrot