University:
Amherst College
Course:
MATH 225 | Fractal Geometry
Academic year:

2019
Views:

303

Pages:

12
Author:

leikfimirw2

Two Big Open Questions 1. Let H = {c ∈ C : fc has an attracting cycle in C}. Big Conjecture: H is dense in M. The density of H would be implied by another Big Conjecture: M is locally connected. 2. What is the area of the boundary ∂M? [Shishikura (1994) showed ∂M has Hausdorff dimension 2.] < « & < M < −.04985 < x < −.04958 and .65044 < y < .65064 < |5 4 s O " O' Where are the c’s with Attracting Fixed Points? Let’s compute, for fc (z) = z 2 + c, the set: H1 = {c ∈ C : fc has an attracting fixed point}. 1. The fixed points are roots of z 2 − z + c = 0. That means √ 1 ± 1 − 4c z= . 2 1 Write 1 − 4c = re iθ , so that c = (1 − re iθ ). (r ≥ 0.) So 4 √ 1 z = (1 ± r e iθ/2 ). 2 2. To be attracting, one must have |fc0 (z)| < 1. That is, |2z| < 1, which is to say √ 1 ± r e iθ/2 < 1. (for at least one choice of + or −.) < (y \ √ 3. Writing e iθ = cos θ + i sin θ, |1 ± r e iθ/2 | < 1 means √ √ θ θ 1 > 1 ± r cos ± i r sin . 2 2 Squaring, that is √ √ θ 2 θ θ 1 > 1 ± r cos + r sin2 = 1 ± 2 r cos + r , 2 2 2 so that √ θ r < ∓2 r cos , 2 for at least one of − or +. 4 4 4 o < \ √ 4. Squaring r < ∓2 r cos(θ/2) gives r 2 < 4r cos2 θ = 2r (1 + cos θ), 2 or in other words, r < 2(1 + cos θ). That means re iθ is inside the cardioid r = 2(1 + cos θ). 1 And that means c = (1 − re iθ ) is inside this cardioid: 4 gp < (y \ ' f) Reminder of The Mandelbrot Set [fp1 < < (y \ ' f) Where are the c’s with Attracting 2-cycles? Let’s compute H2 = {c ∈ C : fc has an attracting 2-cycle}. 1. fc2 (z) = z 4 + 2cz 2 + (c 2 + c), so fc2 (z) − z = z 4 + 2cz 2 − z + (c 2 + c) = (z 2 − z + c)(z 2 + z + (c + 1)). The first factor is the fixed points, so we discard it. Thus, the 2-periodic points are the two roots of z 2 + z + (c + 1) = 0. 4 4 4 o < \ 2. We compute (fc2 )0 (z) = 4z 3 + 4cz = 4zf (z). If z is a 2-periodic point, so that z 2 + z + c + 1 = 0, we get f (z) = z 2 + c = −z − 1, so (fc2 )0 (z) = 4z(−z − 1) = −4(z 2 + z) = 4(c + 1). < « & < M < 3. So fc (z) = z 2 + c has an attracting 2-cycle if |4(c + 1)| < 1. If we write c = a + bi and square, this means (a + 1)2 + b 2 < 2 1 4 , which means c is inside the circle of radius 1/4 centered at −1: O 'o O f < [£? Another Reminder of The Mandelbrot Set [fp 4 y < ( 'f ) \ Periods of Some Other Bulbs The 3-bulb

A Beginner’s Introduction to the Mandelbrot Set Part 3

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