1. Including Complex Dynamics in Complex Analysis Courses
Mandelbrot and Julia sets

2. Benefits: Introduces “modern” topics in mathematics into the course
Illustrates different ways to use
complex analysis “tools”

3. Topics:
Mandelbrot set (complex square roots)
Attracting cycles (Schwarz Lemma)
Understanding chaos (rational maps)
Exponential dynamics (geometry of exp(z))

17. Begin by recalling iteration
Start with a function: 2
x constant
and a seed: x0

18. Then iterate: x = x + constant x = x + constant x = x + constant2
x = x constant 1 2
x = x constant 2 1
Orbit of x 2
x = x constant 3 2 2
x = x constant 4 3
etc.
Goal: understand the fate of orbits.

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

20. Example: x + 0 Seed 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 0 x = 02
Example: x Seed 0
x = 0
x = 0 1
x = 0 2
“A fixed
point”
x = 0 3
x = 0 4
x = 0 5
x = 0 6

21. Example: x - 1 Seed 0 x = 0 x = -1 x = 0 x = -1 x = 0 x = -1 x = 02
Example: x Seed 0
x = 0
x = -1 1
x = 0 2
x = -1
“A two-
cycle” 3
x = 0 4
x = -1 5
x = 0 6

22. Example: x - 1.1 Seed 0 x = 0 x = -1.1 x = 0.11 x = x = x = x =2
Example: x Seed 0
x = 0
x = 1
x = 2
x = 3
time for the
computer!
x = 4
x = 5
x = 6

23. Observation Sometimes orbit of 0 goes to
infinity, other times it does not.

24. Big Question: Answer: How do we understand the chaotic
behavior that occurs?
Answer:
We move to the complex plane
and use tools from complex analysis

25. Complex Iteration 2
Iterate z + c
complex
numbers

26. Example: z + i Seed 0 z = 0 z = i z = -1 + i z = -i z = -1 + i z = -i2
Example: z + i Seed 0
z = 0
z = i 1
z = i 2
z = -i 3
z = i 4
z = -i 5
2-cycle
z = i 6

27. Example: z + 2i Seed 0 z = 0 z = 2i z = -4 + 2i z = 12 - 14i
z = 2i 1
z = i 2
Off to
infinity
z = i 3
z = i 4
z = big 5
z = BIGGER 6

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

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

30. Why use the orbit of 0?
Answer: 0 is a “critical point”
So spend some time talking about
behavior of complex functions near
critical points (not one-to-one) and
near other points (always one-to-one)

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

32. Example: z Seed: In Filled Julia set? Yes 1 Yes -1 Yes i Yes 2i No 5
Yes 1
Yes -1
Yes i
Yes 2i No 5
I doubt it...

33. The filled Julia set of z2 is the unit disk
A little complex analysis:
|z| < 1: orbit of z
|z| > 1: orbit of z infinity
|z| = 1: orbit of z remains on
the unit circle

34. The filled Julia set of z2 is the unit disk
The Julia set is the unit circle

35. Chaos occurs on the Julia set

36. Chaos occurs on the Julia set
3 nearby seeds
Chaos occurs on the Julia set

37. Chaos occurs on the Julia set

38. Chaos occurs on the Julia set

39. Chaos occurs on the Julia set

40. Chaos occurs on the Julia set

41. Chaos occurs on the Julia set

42. Chaos occurs on the Julia set

43. Chaos occurs on the Julia set

44. Chaos occurs on the Julia set

45. Chaos occurs on the Julia set

46. Chaos occurs on the Julia set

47. Chaos occurs on the Julia set

48. Nearby orbits on the Julia set have vastly different behavior

49. i.e., the map just doubles angles
On the Julia set the map is ei e2i
i.e., the map just doubles angles

50. On the Julia set the map is ei e2i
1/5
Rational angles on the Julia
set are eventually periodic.

51. Rational angles on the Julia set are eventually periodic.
1/5
2/5
Rational angles on the Julia
set are eventually periodic.

52. Rational angles on the Julia set are eventually periodic.
1/5
2/5
4/5
Rational angles on the Julia
set are eventually periodic.

53. Rational angles on the Julia set are eventually periodic.
1/5
2/5
3/5
4/5
Rational angles on the Julia
set are eventually periodic.

54. Rational angles on the Julia set are eventually periodic.
1/5
2/5
3/5
4/5
Rational angles on the Julia
set are eventually periodic.

55. Rational angles on the Julia set are eventually periodic.
1/5
2/5
3/5
4/5
Rational angles on the Julia
set are eventually periodic.

56. Rational angles on the Julia set are eventually periodic.
1/5
2/5
3/5
4/5
Rational angles on the Julia
set are eventually periodic.

57. Rational angles on the Julia set are eventually periodic.
1/5
2/5
3/5
4/5
Rational angles on the Julia
set are eventually periodic.

58. But irrational angles on the Julia set are very different.

59. But irrational angles on the Julia set are very different.

60. But irrational angles on the Julia set are very different.

61. But irrational angles on the Julia set are very different.

62. But irrational angles on the Julia set are very different.

63. But irrational angles on the Julia set are very different.

64. But irrational angles on the Julia set are very different.

65. But irrational angles on the Julia set are very different.

66. But irrational angles on the Julia set are very different.

67. Irrational angles have orbits that are dense on the circle.

68. So why do we use the orbit of 0 to plot the Mandelbrot set?
The orbit of the critical point
knows “everything!”

69. So why do we use the orbit of 0 to plot the Mandelbrot set?
The orbit of the critical point
knows “everything!”
To see this, we need the geometry
of complex square roots

70. The square root of a circle (s.c.c.)
that does not surround the origin.... i 1

71. is a pair of s.c.c.’s that do not
The square root of a circle (s.c.c.)
that does not surround the origin.... i
z z2 1
is a pair of s.c.c.’s that do not
surround the origin

72. The square root of a circle (s.c.c.) that touches the origin....1

73. The square root of a circle (s.c.c.) that touches the origin....
z z2 1
is a figure 8 that passes
through the origin

74. The square root of a circle (s.c.c.) that surrounds the origin....1

75. The square root of a circle (s.c.c.) that surrounds the origin....
z z2 1
is one s.c.c. that now
surrounds the origin

76. To find the preimages of a s.c.c.
under z2 + c, we solve:
so that:

77. To find the preimages of a s.c.c.
under z2 + c, we solve:
so that:
So to find the preimage of a s.c.c., we
translate the curve by -c, then take the
complex square root....

78. So now suppose the orbit of 0
escapes to infinity c

79. This large circle is
mapped outside
itself c

80. This large circle is
mapped outside
itself c

81. so all blue
points escape c

82. so all blue
points escape
To find the preimage of
this curve, subtract c
and take the square root... c

83. so all blue
points escape
To find the preimage of
this curve, subtract c
and take the square root...
but subtracting c means
the curve still encircles 0 c

84. so all blue
points escape
so this is the preimage c

85. so all blue
points escape
so these points
also escape c

86. so all blue
points escape
so these points
also escape c

87. so all blue
points escape
so these points
also escape
as do these points c

88. so all blue
points escape
so these points
also escape
as do these points c

89. so all blue
points escape
so these points
also escape
as do these points c

90. 8 so all blue points escape so these points also escape
as do these points c 8

91. Eventually 0 is in a preimage of the
See what’s happening?
Eventually 0 is in a preimage of the
circle, so we get a figure 8 preimage.
And then.....
c = .4975

92. So the filled Julia set consists of infinitely
many distinct components if 0 escapes
(in fact it is a Cantor set).
But when 0 does not escape, we never
get a figure 8, so the filled Julia set
is a connected set.
That’s why the critical point “knows it all.”

93. Attracting cycles
Understanding chaos
Dynamics of the exponential

94. If F is complex analytic and has an
attracting cycle, then there must be
a critical point inside its immediate
basin of attraction.
So F(z) = z2 + c can have at most
one attracting cycle, and the orbit
of 0 must find it.

95. Reason: suppose F has an attracting fixed point p:

96. Then there is a neighborhood U of p mapped 1-1 inside itself.

97. Then there is a neighborhood U of p mapped 1-1 inside itself.
So we have an inverse map
F-1: F(U) U with a
repelling fixed point at p.
F-1 p U

98. Since there are no critical points in the basin, we can continue
F-1 p U

99. Since there are no critical points in
the basin, and on and on .....
F-1 p U

100. Eventually get an open disk V that is
mapped 1-1 to itself by F-1
F-1
... p U V

101. Eventually get an open disk V that is
mapped 1-1 to itself by F-1
F-1 p V

102. But there is a repelling fixed point in V, so this can’t happen by the
Schwarz Lemma
F-1 p V

103. So this begins to describe the structure of the Mandelbrot set:
each “bulb” consists of
parameters for which there is an
attracting cycle of some given period

104. The eventual orbit of 0
Eventual orbit

105. The eventual orbit of 0
Eventual orbit

106. The eventual orbit of 0
3-cycle

107. The eventual orbit of 0
3-cycle

108. The eventual orbit of 0
3-cycle

109. The eventual orbit of 0
3-cycle

110. The eventual orbit of 0

111. The eventual orbit of 0

112. The eventual orbit of 0
4-cycle

113. The eventual orbit of 0
4-cycle

114. The eventual orbit of 0
4-cycle

115. The eventual orbit of 0
4-cycle

116. The eventual orbit of 0
4-cycle

117. The eventual orbit of 0

118. The eventual orbit of 0

119. The eventual orbit of 0
5-cycle

120. The eventual orbit of 0
5-cycle

121. The eventual orbit of 0
5-cycle

122. The eventual orbit of 0
5-cycle

123. The eventual orbit of 0
5-cycle

124. The eventual orbit of 0
5-cycle

125. The eventual orbit of 0
5-cycle

126. The eventual orbit of 0
2-cycle

127. The eventual orbit of 0
2-cycle

128. The eventual orbit of 0
2-cycle

129. The eventual orbit of 0
fixed point

130. The eventual orbit of 0
fixed point

131. The eventual orbit of 0
fixed point

132. How understand the
periods of the bulbs?

133. How understand the
periods of the bulbs?

134. junction point
three spokes
attached

135. junction point
three spokes
attached
Period 3 bulb

138. Period 4 bulb

141. Period 5 bulb

144. Period 7 bulb

148. Period 13 bulb

149. Complex exponential dynamics
Let E (z) = exp(z) where > 0.
Differences: no critical points, but one
“asymptotic value” at z = 0.

150. Complex exponential dynamics
Let E (z) = exp(z) where > 0.
Differences: no critical points, but one
“asymptotic value” at z = 0.
H is wrapped
infinitely often
around 0, which
plays the role of
the critical value H

151. Complex exponential dynamics
Let E (z) = exp(z) where > 0.
Differences: no critical points, but one
“asymptotic value” at z = 0.
The Julia set is now the closure of the
set of escaping points, not the
boundary of this set

152. When 0 < < 1/e, the Julia set is
a “Cantor bouquet,” i.e., infinitely
many curves extending to infinity in
the right half plane.

153. 0 < < 1/e

154. attracting
fixed point q

155. q p
repelling
fixed point

156. q x0 p

157. So where is J?

158. So where is J?

159. So where is J?
Green points lie in the basin of q, so not in the Julia set

160. So where is J?
Green points lie in the basin of q, so not in the Julia set

161. So where is J?
Green points lie in the basin of q, so not in the Julia set

162. So where is J?
Green points lie in the basin of q, so not in the Julia set

163. So where is J?
Green points lie in the basin of q, so not in the Julia set

164. The Julia set is a collection of curves (hairs) in the right half plane, each with an endpoint
and a stem.
hairs
endpoints
stems

165. A “Cantor bouquet” q p

166. Colored points escape to and so are in the Julia set.q p

171. When = 1/e, E undergoes a “saddle node bifurcation”
but much more happens...

172. When = 1/e, E undergoes a “saddle node bifurcation”
The two fixed points on the real line disappear and
the orbit of the asymptotic value now escapes....

173. And, because of this, the Julia instantaneously
becomes the entire complex plane!

397. Replay

398. How understand the chaotic behavior?z2
z i

399. so use that information
How understand the chaotic behavior?
We understand what’s
going on here,
so use that information
to understand what’s
going on here.

400. How understand the chaotic behavior?
These angles are
just rotated by z2

401. How understand the chaotic behavior?
These angles are
just rotated by z2
So we can find a “conjugacy”
that creates angles here
that map in the same way

402. Example: z2 - 2 z2 z2 - 2 How to map the exterior of the
circle to the exterior of [-2,2]? -2 2 z2
z2 - 2

403. Consider H(z) = z + 1/z -2 2 z2
z2 - 2

404. Consider H(z) = z + 1/z
H takes rays to
hyperbolas -2 2 z2
z2 - 2

405. Consider H(z) = z + 1/z
Well, usually
that’s the case -2 2 z2
z2 - 2

406. Consider H(z) = z + 1/z z2 z2 - 2 And H takes the circle to [-2, 2] -2

407. Consider H(z) = z + 1/z z2 z2 - 2 And H takes the circle to [-2, 2]
since eit + e-it = 2 cos(t) -2 2 z2
z2 - 2

408. On the circle we have: H H
On [-2,2]
we have:
What is this function???

409. On the circle we have: H H

410. On the circle we have: H H

411. On the circle we have: H H
So this function is w w2 - 2

412. So the orbits of F(z) = z2 on the unit circle (which
we completely understand are taken to orbits of
G(z) = z by the “conjugacy” H(z) = z + 1/z. z
F(z)
F2(z)
F3(z)
F4(z) ... H H H H H w
G(w)
G2(w)
G2(w)
G4(z) ...
And that’s how we understand the chaotic behavior
on the Julia sets.