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Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z)) 3. Rational maps (z n + /z n )

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We’ll investigate chaotic behavior in the dynamical plane (the Julia sets) z 2 + c exp(z) z 2 + /z 2

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As well as the structure of the parameter planes. z 2 + c exp(z)z 3 + /z 3 (the Mandelbrot set)

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A couple of subthemes: 1.Some “crazy” mathematics 2.Great undergrad research topics

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The Fractal Geometry of the Mandelbrot Set

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How to count The Fractal Geometry of the Mandelbrot Set

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The Fractal Geometry of the Mandelbrot Set How to add How to count

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Many people know the pretty pictures...

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but few know the even prettier mathematics.

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Oh, that's nothing but the 3/4 bulb....

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...hanging off the period 16 M-set.....

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...lying in the 1/7 antenna...

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...attached to the 1/3 bulb...

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...hanging off the 3/7 bulb...

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...on the northwest side of the main cardioid.

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Oh, that's nothing but the 3/4 bulb, hanging off the period 16 M-set, lying in the 1/7 antenna of the 1/3 bulb attached to the 3/7 bulb on the northwest side of the main cardioid.

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Start with a function: x + constant 2

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Start with a function: x + constant 2 and a seed: x 0

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Then iterate: x = x + constant 10 2

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Then iterate: x = x + constant

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Then iterate: x = x + constant

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Then iterate: x = x + constant

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Then iterate: x = x + constant Orbit of x 0 etc. Goal: understand the fate of orbits.

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Example: x + 1 Seed 0 2 x = 0 0 x =

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Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x =

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Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x =

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Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 4 5 6

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Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = 5 6

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Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = 6

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Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = BIGGER 6

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Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = BIGGER 6 “Orbit tends to infinity”

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Example: x + 0 Seed 0 2 x = 0 0 x =

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Example: x + 0 Seed 0 2 x = x =

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Example: x + 0 Seed 0 2 x = x =

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Example: x + 0 Seed 0 2 x = x = 4 5 6

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Example: x + 0 Seed 0 2 x = “A fixed point”

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Example: x - 1 Seed 0 2 x = 0 0 x =

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Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x =

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Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x =

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Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 4 5 6

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Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 0 4 x = 5 6

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Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 0 4 x = -1 5 x = 0 6 “A two- cycle”

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Example: x Seed 0 2 x = 0 0 x =

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Example: x Seed 0 2 x = 0 0 x = x =

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Example: x Seed 0 2 x = 0 0 x = x = x =

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Example: x Seed 0 2 x = 0 0 x = x = x = time for the computer!

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Observation: For some real values of c, the orbit of 0 goes to infinity, but for other values, the orbit of 0 does not escape.

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Complex Iteration Iterate z + c 2 complex numbers

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Example: z + i Seed 0 2 z = 0 0 z =

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Example: z + i Seed 0 2 z = 0 0 z = i 1 z =

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Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z =

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Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = 4 5 6

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Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = 5 6

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Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = -i 5 z = 6

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Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = -i 5 z = -1 + i 6 2-cycle

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Example: z + i Seed i -i

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Example: z + i Seed i -i

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Example: z + i Seed i -i

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Example: z + i Seed 0 2 -i 1 i

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Example: z + i Seed i -i

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Example: z + i Seed 0 2 -i 1 i

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Example: z + i Seed i -i

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Example: z + i Seed 0 2 -i 1 i

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Example: z + 2i Seed 0 2 z = 0 0 z =

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Example: z + 2i Seed 0 2 z = 0 0 z = 2i 1 z = i 2 z = i 3 z = i 4 z = big 5 z = BIGGER 6 Off to infinity

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Same observation Sometimes orbit of 0 goes to infinity, other times it does not.

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The Mandelbrot Set: All c-values for which orbit of 0 does NOT go to infinity. Why do we care about the orbit of 0?

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The Mandelbrot Set: All c-values for which orbit of 0 does NOT go to infinity. As we shall see, the orbit of the critical point determines just about everything for z 2 + c. 0 is the critical point of z 2 + c.

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Algorithm for computing M Start with a grid of complex numbers

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Algorithm for computing M Each grid point is a complex c-value.

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Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. red = fastest escape

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Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. orange = slower

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Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. yellow green blue violet

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Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 does not escape, leave that grid point black.

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Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 does not escape, leave that grid point black.

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The eventual orbit of 0

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3-cycle

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The eventual orbit of 0 3-cycle

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The eventual orbit of 0 3-cycle

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The eventual orbit of 0 3-cycle

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The eventual orbit of 0 3-cycle

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The eventual orbit of 0 3-cycle

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The eventual orbit of 0 3-cycle

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The eventual orbit of 0 3-cycle

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The eventual orbit of 0 3-cycle

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The eventual orbit of 0

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4-cycle

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The eventual orbit of 0 4-cycle

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The eventual orbit of 0 4-cycle

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The eventual orbit of 0 4-cycle

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The eventual orbit of 0 4-cycle

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The eventual orbit of 0 4-cycle

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The eventual orbit of 0 4-cycle

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The eventual orbit of 0 4-cycle

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The eventual orbit of 0

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5-cycle

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The eventual orbit of 0 5-cycle

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The eventual orbit of 0 5-cycle

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The eventual orbit of 0 5-cycle

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The eventual orbit of 0 5-cycle

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The eventual orbit of 0 5-cycle

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The eventual orbit of 0 5-cycle

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The eventual orbit of 0 5-cycle

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The eventual orbit of 0 5-cycle

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The eventual orbit of 0 5-cycle

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The eventual orbit of 0 5-cycle

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The eventual orbit of 0 2-cycle

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The eventual orbit of 0 2-cycle

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The eventual orbit of 0 2-cycle

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The eventual orbit of 0 2-cycle

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The eventual orbit of 0 2-cycle

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The eventual orbit of 0 fixed point

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The eventual orbit of 0 fixed point

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The eventual orbit of 0 fixed point

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The eventual orbit of 0 fixed point

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The eventual orbit of 0 fixed point

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The eventual orbit of 0 fixed point

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The eventual orbit of 0 fixed point

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The eventual orbit of 0 fixed point

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The eventual orbit of 0 goes to infinity

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The eventual orbit of 0 goes to infinity

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The eventual orbit of 0 goes to infinity

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The eventual orbit of 0 goes to infinity

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The eventual orbit of 0 goes to infinity

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The eventual orbit of 0 goes to infinity

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The eventual orbit of 0 goes to infinity

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The eventual orbit of 0 goes to infinity

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The eventual orbit of 0 goes to infinity

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The eventual orbit of 0 goes to infinity

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The eventual orbit of 0 goes to infinity

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The eventual orbit of 0 gone to infinity

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One reason for the importance of the critical orbit: If there is an attracting cycle for z 2 + c, then the orbit of 0 must tend to it.

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How understand the of the bulbs? periods

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How understand the of the bulbs? periods

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junction point three spokes attached

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Period 3 bulb junction point three spokes attached

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Period 4 bulb

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Period 5 bulb

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Period 7 bulb

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Period 13 bulb

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Filled Julia Set:

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Fix a c-value. The filled Julia set is all of the complex seeds whose orbits do NOT go to infinity.

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Example: z 2 Seed: 0 In filled Julia set?

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Example: z 2 Seed: 0Yes In filled Julia set?

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Example: z 2 Seed: 0Yes 1 In filled Julia set?

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Example: z 2 Seed: 0Yes 1 In filled Julia set?

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Example: z 2 Seed: 0Yes 1 In filled Julia set?

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Example: z 2 Seed: 0Yes 1 Yes In filled Julia set?

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Example: z 2 Seed: 0Yes 1 Yes i In filled Julia set?

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Example: z 2 Seed: 0Yes 1 Yes i In filled Julia set?

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Example: z 2 Seed: 0Yes 1 Yes i 2i In filled Julia set?

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Example: z 2 Seed: 0Yes 1 Yes i 2i No In filled Julia set?

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Example: z 2 Seed: 0Yes 1 Yes i 2i No 5 In filled Julia set?

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Example: z 2 Seed: 0Yes 1 Yes i 2i No 5No way In filled Julia set?

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Filled Julia Set for z 2 All seeds on and inside the unit circle. i 1

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The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic”

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The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

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The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

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The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

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The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

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The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

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The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

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The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

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The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

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The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

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The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

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Other filled Julia sets

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c = 0

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Other filled Julia sets c = -1

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Other filled Julia sets c = -1

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Other filled Julia sets c = -1

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Other filled Julia sets c = -1

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Other filled Julia sets c = -1

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Other filled Julia sets c = -1

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Other filled Julia sets c = -1

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Other filled Julia sets c = -1

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Other filled Julia sets c = i

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Other filled Julia sets c = i

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Other filled Julia sets c = i

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Other filled Julia sets c = i

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Other filled Julia sets c = i

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Other filled Julia sets c = i

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If c is in the Mandelbrot set, then the filled Julia set is always a connected set.

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Other filled Julia sets But if c is not in the Mandelbrot set, then the filled Julia set is totally disconnected.

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Other filled Julia sets c =.3

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Other filled Julia sets c =.3

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Other filled Julia sets c =.3

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Other filled Julia sets c =.3

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Other filled Julia sets c =.3

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Other filled Julia sets c = i

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Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c:

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Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c: If the orbit of 0 goes to infinity, the Julia set is a Cantor set (totally disconnected, “fractal dust,” a scatter of uncountably many points.

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Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c: But if the orbit of 0 does not go to infinity, the Julia set is connected (just one piece). If the orbit of 0 goes to infinity, the Julia set is a Cantor set (totally disconnected, “fractal dust,” a scatter of uncountably many points.

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Animations: In and out of M arrangement of the bulbs Saddle node Period doubling Period 4 bifurcation

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How do we understand the arrangement of the bulbs?

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How do we understand the arrangement of the bulbs? Assign a fraction p/q to each bulb hanging off the main cardioid. q = period of the bulb

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Where is the smallest spoke in relation to the “principal spoke”? p/3 bulb principal spoke

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1/3 bulb principal spoke The smallest spoke is located 1/3 of a turn in the counterclockwise direction from the principal spoke.

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1/3 bulb 1/3

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1/3 bulb 1/3

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1/3 bulb 1/3

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1/3 bulb 1/3

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1/3 bulb 1/3

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1/3 bulb 1/3

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1/3 bulb 1/3

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1/3 bulb 1/3

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1/3 bulb 1/3

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1/3 bulb 1/3

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??? bulb 1/3

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1/4 bulb 1/3

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1/4 bulb 1/3 1/4

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1/4 bulb 1/3 1/4

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1/4 bulb 1/3 1/4

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1/4 bulb 1/3 1/4

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1/4 bulb 1/3 1/4

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1/4 bulb 1/3 1/4

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1/4 bulb 1/3 1/4

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1/4 bulb 1/3 1/4

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1/4 bulb 1/3 1/4

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??? bulb 1/3 1/4

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2/5 bulb 1/3 1/4

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2/5 bulb 1/3 1/4 2/5

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2/5 bulb 1/3 1/4 2/5

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2/5 bulb 1/3 1/4 2/5

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2/5 bulb 1/3 1/4 2/5

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2/5 bulb 1/3 1/4 2/5

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??? bulb 1/3 1/4 2/5

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3/7 bulb 1/3 1/4 2/5

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3/7 bulb 1/3 1/4 2/5 3/7

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3/7 bulb 1/3 1/4 2/5 3/7

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3/7 bulb 1/3 1/4 2/5 3/7

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3/7 bulb 1/3 1/4 2/5 3/7

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3/7 bulb 1/3 1/4 2/5 3/7

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3/7 bulb 1/3 1/4 2/5 3/7

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3/7 bulb 1/3 1/4 3/7 2/5

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??? bulb 1/3 1/4 3/7 2/5

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1/2 bulb 1/3 1/4 3/7 1/2 2/5

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1/2 bulb 1/3 1/4 3/7 1/2 2/5

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1/2 bulb 1/3 1/4 3/7 1/2 2/5

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1/2 bulb 1/3 1/4 3/7 1/2 2/5

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??? bulb 1/3 1/4 3/7 1/2 2/5

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2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

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2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

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2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

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2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

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2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

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2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

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How to count

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1/4 How to count

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1/3 1/4 How to count

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1/3 1/4 2/5 How to count

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1/3 1/4 2/5 3/7 How to count

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1/3 1/4 2/5 3/7 1/2 How to count

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1/3 1/4 2/5 3/7 1/2 2/3 How to count

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1/3 1/4 2/5 3/7 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers. How to count

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1/3 1/4 2/5 3/7 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers. 1/101 32,123/96,787 How to count

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Animations: Mandelbulbs Spiralling fingers

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How to add

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1/2

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How to add 1/2 1/3

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How to add 1/2 1/3 2/5

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How to add 1/2 1/3 2/5 3/7

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+= 1/2 + 1/3 = 2/5

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+= 1/2 + 2/5 = 3/7

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22 1/2 0/1 Here’s an interesting sequence:

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22 1/2 0/1 Watch the denominators 1/3

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22 1/2 0/1 Watch the denominators 1/3 2/5

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22 1/2 0/1 Watch the denominators 1/3 2/5 3/8

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22 1/2 0/1 Watch the denominators 1/3 2/5 3/8 5/13

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22 1/2 0/1 What’s next? 1/3 2/5 3/8 5/13

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22 1/2 0/1 What’s next? 1/3 2/5 3/8 5/13 8/21

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22 1/2 0/1 The Fibonacci sequence 1/3 2/5 3/8 5/13 8/21 13/34

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The Farey Tree

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How get the fraction in between with the smallest denominator?

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The Farey Tree How get the fraction in between with the smallest denominator? Farey addition

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The Farey Tree

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.... essentially the golden number

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Another sequence (denominators only) 1 2

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Another sequence (denominators only) 1 2 3

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Another sequence (denominators only)

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Another sequence (denominators only)

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Another sequence (denominators only)

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Another sequence (denominators only)

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sequence Devaney

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The Dynamical Systems and Technology Project at Boston University website: math.bu.edu/DYSYS: Have fun! Mandelbrot set explorer; Applets for investigating M-set; Applets for other complex functions; Chaos games, orbit diagrams, etc.

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Farey.qt Farey tree D-sequence Continued fraction expansion Far from rationals Other topics Website

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Continued fraction expansion Let’s rewrite the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,..... as a continued fraction:

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Continued fraction expansion 1212 = 1212 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

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Continued fraction expansion 1313 = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

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Continued fraction expansion 2525 = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

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Continued fraction expansion 3838 = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

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Continued fraction expansion = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

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Continued fraction expansion = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

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Continued fraction expansion = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

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Continued fraction expansion = essentially the 1/golden number the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

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We understand what happens for = 1a1a + 1b1b + 1c1c 1d1d + 1e1e + 1f1f + 1g1g + where all entries in the sequence a, b, c, d,.... are bounded above. But if that sequence grows too quickly, we’re in trouble!!! etc.

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The real way to prove all this: Need to measure: the size of bulbs the length of spokes the size of the “ears.”

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There is an external Riemann map : C - D C - M taking the exterior of the unit disk to the exterior of the Mandelbrot set.

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takes straight rays in C - D to the “external rays” in C - M 01/2 1/3 2/3 external ray of angle 1/3

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Suppose p/q is periodic of period k under doubling mod 1: period 2 period 3 period 4

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Suppose p/q is periodic of period k under doubling mod 1: period 2 period 3 period 4 Then the external ray of angle p/q lands at the “root point” of a period k bulb in the Mandelbrot set.

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0 1/3 2/3 0 is fixed under angle doubling, so lands at the cusp of the main cardioid.

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0 1/3 2/3 1/3 and 2/3 have period 2 under doubling, so and land at the root of the period 2 bulb. 2

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0 1/3 2/3 And if lies between 1/3 and 2/3, then lies between and. 2

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0 1/3 2/3 So the size of the period 2 bulb is, by definition, the length of the set of rays between the root point rays, i.e., 2/3-1/3=1/3. 2

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0 1/3 2/3 1/15 and 2/15 have period 4, and are smaller than 1/ /7 2/7 3/7 4/7 5/7 6/ /15 2/15

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0 1/3 2/3 1/15 and 2/15 have period 4, and are smaller than 1/ /7 2/7 3/7 4/7 5/7 6/ /15 2/15

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0 1/3 2/3 1/7 2/7 3/7 4/7 5/7 6/ /15 2/15 3/15 and 4/15 have period 4, and are between 1/7 and 2/7....

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0 1/3 2/3 3/15 and 4/15 have period 4, and are between 1/7 and 2/ /7 2/7 3/7 4/7 5/7 6/ /15 2/15

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1/72/7 3/15 and 4/15 have period 4, and are between 1/7 and 2/7....

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1/72/7 3/15 and 4/15 have period 4, and are between 1/7 and 2/ /154/15

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So what do we know about M? All rational external rays land at a single point in M.

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So what do we know about M? All rational external rays land at a single point in M. Rays that are periodic under doubling land at root points of a bulb. Non-periodic rational rays land at Misiurewicz points (how we measure length of antennas).

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So what do we know about M? “Highly irrational” rays also land at unique points, and we understand what goes on here. “Highly irrational" = “far” from rationals, i.e.,

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So what do we NOT know about M? But we don't know if irrationals that are “close” to rationals land. So we won't understand quadratic functions until we figure this out.

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MLC Conjecture: The boundary of the M-set is “locally connected” --- if so, all rays land and we are in heaven!. But if not......

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The Dynamical Systems and Technology Project at Boston University website: math.bu.edu/DYSYS Have fun!

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A number is far from the rationals if:

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This happens if the “continued fraction expansion” of has only bounded terms.

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