Complex Numbers

푴풂풏풅풆풍풃풓풐풕 푺풆풕

Julia set of Quadratic Polynomials

Myong-Hi Kim SUNY Old Westbury for math club This presentation contains parts of the following sources From 풀풐풖풕풖풃풆. 풄풐풎 Dimension 5, 6 Lecture movies by Adrien Douady From Prof. Bob Devaney’s site Boston University http://math.bu.edu/DYSYS/FRACGEOM/index.html For slides, their slides are prettier than mine. So I am borrowing Bob’s slide from his website. XavierBuff&ArnaudCh´eritat Universit´ePaulSabatier(ToulouseIII) Before the talk starts 1. Part of Dimension Ep. 5 complex numbers I https://www.youtube.com/watch?v=2kbM96Jr4nk 2. For 2 min from 8.50 min of Dimensions Ep 6 https://www.youtube.com/watch?v=XzT5XSgkLvk From 8 min 50 second

After the talk we continue watch Ep. 6 to the end https://www.youtube.com/watch?v=onMLujxxwug In the usual real plane 푅2, Add component wise and

Give life of algebra by : 푖2 = −1 Arithmetic in complex plane C

Consider 푖 as a variable in college algebra Whenever you see 풊ퟐ Substitute 풊ퟐ by -1. 풂 + 풃풊 + 풄 + 풅풊 = 풂 + 풄 + 풄 + 풅 풊

For example, ퟐ + 풊 + ퟑ + ퟕ풊 = ퟓ + ퟖ풊 (ퟐ + 풊) + (ퟑ + ퟓ풊)

= ퟐ + ퟑ + ퟓ풊

= ퟓ + ퟖ풊 풊ퟑ, 풊ퟒ

풊ퟑ = 풊ퟐ ⋅ 풊

= −ퟏ 풊

= −풊

ퟐ 퐢ퟒ = 풊ퟐ = −ퟏ ퟐ = ퟏ 푬풙풂풎풑풍풆 (FOIL) ퟐ + 풊 ⋅ (ퟑ + ퟓ풊)

= ퟐ ퟑ + ퟓ풊 + 풊(ퟑ + ퟓ풊)

= (ퟐ ⋅ ퟑ + ퟐ ⋅ ퟓ풊) + (풊 ⋅ ퟑ + 풊 ⋅ ퟓ ⋅ 풊)

= (ퟔ + ퟏퟎ풊) + ( ퟑ ⋅ 풊 + ퟓ ⋅ 풊 ⋅ 풊)

= (ퟔ + ퟏퟎ풊) + ( ퟑ ⋅ 풊 + ퟓ(−ퟏ) = (ퟔ − ퟓ) + ( ퟏퟎ풊 + ퟑ ⋅ 풊 ) = ퟏ + ퟏퟑ풊 Beauty of Multiplication of two complex numbers •Radius (length ) are multiplied •Angles are added. •Richard Feynman once said the most beautiful formula … Euler’s formula: 푖푎 푖푏 푖 푎+푏 푟1푒 ⋅ 푟2푒 = 푟1푟2푒 writing cos 푎 + 푖 sin 푏 as 푒푖푎 Beauty of Multiplication of two complex numbers

When a complex number 푧 = 푟((cos 휃) + 푖 sin 휃) is squared, Radius is squared and angle is doubled. Notice that the quarter circle with radius 2 is mapped to the half circle with radius 4. The algebra 풊ퟐ = −ퟏ brings beautiful and rich

structure

to mathematics Iterates of a function: repeated Composition by itself

풇: 푪 → 푪 풇 ∘ 풇 풛 = 풇 풇 풛 (풇 ∘ 풇 ∘ 풇)(풛) = 풇(풇 ∘ 풇 풛 ) = 풇(풇(풇(풛)))

Write iterates using power 풇 ퟏ 풛 = 풇 풛 풇 풏 풛 = 풇 풇 풏−ퟏ 풛 = 풇 풇 풇 ⋯ 풛 풇: 푪 → 푪 풇 풛 = 풛ퟐ − ퟐ

풇 ퟏ ퟎ = 풇 ퟎ = −ퟐ

풇 ퟐ ퟎ = 풇 풇 ퟏ ퟎ = 풇 −ퟐ = (−ퟐ)ퟐ−ퟐ = ퟐ

풇 ퟑ ퟎ = 풇 ퟐ = ퟐퟐ − ퟐ = ퟐ

풇 ퟒ ퟎ , = ퟐ ⋯ , Iterates of 풇 풏 ퟎ , 풇 풛 = 풛ퟐ − ퟐ 풇 ퟏ ퟎ 풇 ퟎ ퟎퟐ − ퟐ −ퟐ

ퟐ 풇 ퟎ 풇 풇 ퟏ ퟎ 풇(−ퟐ) ퟐ

ퟑ 풇 ퟎ 풇 풇 ퟐ ퟎ 풇(ퟐ) ퟐ

풇 ퟒ ퟎ 풇(ퟐ) ퟐ

⋮ ⋮ ⋮ 푫풆풇풊풏풊풕풊풐풏: fixed point

z is a fixed point of 풇 풛 iff z=퐟 퐳

풛 = ퟐ is a fixed point of 풇 풛 = 풛ퟐ − ퟐ

ퟎ and −ퟐ are preimages of a fixed point of 풛ퟐ − ퟐ. More example: 풇 풛 = 풛ퟐ + 풊

풇 ퟏ ퟎ = 풇 ퟎ = ퟎퟐ + 풊 = 풊

풇ퟐ ퟎ = 풇 풇(ퟎ) = 풊ퟐ + 풊 = −ퟏ + 풊

풇ퟑ ퟎ = 풇 풇 ퟐ ퟎ = 풇 −ퟏ + 풊 = −ퟏ + 풊 ퟐ + 풊 = ( −ퟏ ퟐ − ퟐ풊 + 풊 ퟐ + 풊

= ퟏ − ퟐ풊 − ퟏ + 풊=-i 풇 풛 = 풛ퟐ + 풊 Every two periods -1+i and i are repeated!!!

• 풇 ퟎ = 풊 • 풇 ퟐ ퟎ = −ퟏ + 풊 • 풇 ퟑ ퟎ = −풊 • 풇 ퟒ ퟎ = −ퟏ + 풊 • 풇 ퟓ (ퟎ) = −풊 • 풇 ퟔ ퟎ = −ퟏ + 풊 • 풇 ퟕ ퟎ = −풊 • ⋯ • −ퟏ + 풊 and −풊 are called periodic points with period 2 of the polynomial 풇 풛 = 풛ퟐ + 풊

• 풊 is preperiodic: it is a preimage of a periodic point. Definition: Periodic point , pre-periodic point

•풛 is a periodic point of 풇 if 풇 풌 풛 = 풛 •The smallest positive number 풌 is called the period of 풛 •Preimages of periodic point is called pre-periodic point. • F퐨퐫 풇 풛 = 풛ퟐ + 풊 • −ퟏ + 풊 and −풊 are periodic points with period 2 • 풊 is preperiodic In the movie, the red set is called Mandelbrot set and the red set is called the filled in Julia set of 푧2 + 푐, where c is the point where the black dot in the red set is pointing. Definition

Mandelbrot set M Denote 풏 ퟐ 풇풄 = 풛 + 풄 풏 푴 = 풄 ∈ C |풇풄 ퟎ | is bounded}

It is known that 풏 풄 ∈ 푴 풊풇풇 풍풊풎풔풖풑풏→∞ |풇풄 ퟎ | ≤ ퟐ

How to check if it is in M?

• Pick a point c • 푓 푧 = 푧2 + 푐 푛 • Compute 푧푛 = 푓 0 , 푛 = 1,2,3, ⋯ • If 푧푛 does not diverges to infinity then it is in M. How to make Mandelbrot set? How to check if it is in M?

c=0 is in M c=4 is not in M z0 = 0 z0 = 0 2 2 z1 = z0 + 0 = 0 z1 = z0 + 4 = 4 2 2 2 z2 = z1 + 0 z2 = 4 + 4 = 20 > 4 2 2 n = 0 + 0 = 0 zn+1 = zn + c > 4 ⋯ Conclude zn = 0 {|zn |} is not bounded And |z | = 0 for all n n So, c=4 is not in M So c=0 is in M How does a computer plot Mandelbrot set How does a computer plot Mandelbrot set

•Density plot in the range: [-2, 0.55]x[-1,5, 1.5] •Choose a number of grid points, say 200x200. •Use every grid point as constant value c of f •Set 푓 푧 = 푧2 + 푐 푛 •Compute 푧푛 = 푓 0 , 푛 = 1,2,3, ⋯ , 100 •If |푧푛| < 2 plot the grid point c in M. •Else go to the next grid. Mandelbrot set 푴풂풊풏 푩풐풅풚 풐풇 푴 Cardioid is the locust of 풆ퟐ흅풊풕 풆ퟐ흅풊풕 푪(풕) = ퟏ − ퟐ ퟐ 풑 풑 ퟐ흅 풊 ퟐ흅 풊 풑 풆 풒 풆 풒 Bulbs are attached at points, 푪 = ퟏ − 풒 ퟐ ퟐ Notice that three are rational number many bulb points on the main cardioid 풑 풑 ퟐ흅 풊 ퟐ흅 풊 풆 풒 풆 풒 푪풑 = ퟏ − 풒 ퟐ ퟐ 푪풓풂풛풚 풃풖풕 푰풏풕풆풓풆풔풕풊풏품

푺풆풍풇 − 푺풊풎풊풍풂풓

M is Connected

We don′t know if M is locally connectedness. is still open In the sequel, Zoom zoom

•We will zoom where drawn by squares

3 bulb 4 1 16 antenna 3 bulb 7 Julia sets How it looks? Relation between M and J and FJ Definitions of filled Julia set and Julia set What is a Julia set and how to plot them?

푧2 + 푐, where c is the point where the black dot in the red set is pointing. 풛ퟐ+c Definitions

푭풊풍풍풆풅 푱풖풍풊풂 = 퐳 ∈ 푪 풇 풏 ퟎ → ∞}

Julia set is the boundary of Filled Julia Set

Julia set is the closure of repelling periodic points Fix a complex number c consider a quadratic polynomial 풇 풛 = 풛ퟐ+c

For every 풛 ∈ 푪, Compute iterates of 풇(풛) Check 풇 풏 풛 → ∞ Maple 푐 = −1.037 + 0.17푖 푓 푧 = 푧2-1.037 + 0.17*퐼

• Fix a complex number c consider a quadratic polynomial 풇 풛 = 풛ퟐ+c

For every 풛 ∈ 푪, Compute iterates of 풇(풛) Check 풇 풏 풛 → ∞ Most Julia set we see has Lebesgue measure (area) 0.

Picture we see are filled Julia set whose boundary is the Julia set. To visualize Julia set we need to plot filled Julia set.

It is open question if there is a Julia set of quadratic with positive Lebesgue measure.

Fractal dimension (Hausdorff dimension) is invented Relation between Julia set of 풛ퟐ + 풄 풂풏풅 where in Mandelbrot set 푴풂풊풏 푩풐풅풚 풐풇 푴 Cardioid Bulbs are attached at points 푪풑 are interesting. 풒 periodic points of 풛ퟐ + 풄 with period q

풑 풑 ퟐ흅 풊 ퟐ흅 풊 풆 풒 풆 풒 푪풑 = ퟏ − 풒 ퟐ ퟐ 푴풂풊풏 푩풐풅풚 풐풇 푴 Cardioid

풑 풑 ퟐ흅 풊 ퟐ흅 풊 풆 풒 풆 풒 푪풑 = ퟏ − 풒 ퟐ ퟐ The slide is from “Polynomial Juliasets with positive measure” by XavierBuff&ArnaudCh´eritat Universit´ePaulSabatier(ToulouseIII) presented in memory of Adien Douday in Theorem (Fatou Julia, 1919): Julia set of 풇: 푪 → 푪 is

Either connected

or Totally disconnected

Points c Outside of Mandelbrot set, Filled Julia set 풛ퟐ + 풄 is totally disconnected like dusts Julia set of 풛ퟐ + 풄, with c in Bulb of Mandelbrot set 푴풂풊풏 푩풐풅풚 풐풇 푴 Cardioid has bulbs at 푪풑 풒 ퟐ Julia set of 풛 + 푪풑 풒 has periodic points with period q

풑 풑 ퟐ흅 풊 ퟐ흅 풊 풆 풒 풆 풒 푪풑 = ퟏ − 풒 ퟐ ퟐ 퐽푢푙푖푎 푠푒푡 표푓 푧2 + 0 is the unit circle. At golden ratio c a + b a = = ϕ a b 1 + 5 ϕ = ≈ 1.618 … 2 Julia set of 푧2 + 휙 푱풖풍풊풂 풔풆풕 풐풇 풇 풛 = 풛ퟐ−1 -1 is at the bulb C(1/2) In the north bulb at 1/3 bulb period 3 Douady’s rabbit Julia set of 푧2 + 푐, c=-0.125+ 0.731 I, c~-0.12256+0.74486i

From Wikipedia Maple produced Julia set Rabbits at c=-0.12+0.75I Julia set for fc, c=(φ−2)+(φ−1)i =-0.4+0.6i Getting chaotic Chaos !!!! Julia set for filled Julia sets that is not connected is similar to the Cantor Middle Thirds Set. 푴풐풔풕 푱풖풍풊풂 풔풆풕 풘풆 풔풆풆 풉풂풗풆 풎풆풂sure (area) 0.

Julia set of 풛ퟐ+c with c on the boundary of Mandelbrot set are 퐦퐨퐫퐞 interesting points , starting to be chaotic!!!

Cantor set: Totally disconnected like dusts 푹풆풄풆풏풕풍풚, found Julia sets with postive measure. More crazy/interesting points Misiurewicz points Misiurewicz points Are Points c in M, for which 0 is preperiodic of 풛ퟐ + 풄 . Notice that 0 is the only critical point of 풛ퟐ + 풄. Misiurewicz points

•Dense in the boundary of M •Filled Julia set is equal to Julia set. •Filled Julia set has no interior •Mandelbrot set and Julia set are asymptotically similar. •It disconnect M at least into 3 components M-set: c= 0.4244 + 0.200759i From Wikipdia Misiurewicz points M21 = −2, M22 = i, 퐶 = 푀23 https://en.wikipedia.org/wiki/Misiurewicz_point Julia set of 풇 풛 = 풛ퟐ + ퟏ is totally disconnected

Has a Lebesgue measure 0

OK to end here Or go more Hausdorff Dimension?

푷풂풓풕풊풕풊풐풏 풐풇 풂 풔풆풕 풊풏풕풐 푵 풄풆풍풍풔 with cells with diameter 풅 휹 such that 푵 풅풊풂풎풆풕풆풓 휹 = ퟏ Example: Cantor set 풍풏ퟐ Haussdorff dimension 풍풏 ퟑ in n step 푵 = ퟐ풏 휹 풏 ퟏ ퟐ풏 = ퟏ ퟑ

풏 ퟐ 풍풏ퟐ → = ퟏ → 휹 = ퟑ휹 풍풏 ퟑ Example: Cantor set Looks like dusts

Totally disconnected Watch Movie

• Thank you