1. Singular Perturbations of Complex Polynomials
Dynamics of the family of complex maps
with:
Paul Blanchard
Toni Garijo
Matt Holzer
Dan Look
Sebastian Marotta
Mark Morabito
Monica Moreno Rocha
Kevin Pilgrim
Elizabeth Russell
Yakov Shapiro
David Uminsky
Sum Wun Ellce

2. Singular Perturbations of Complex Polynomials
1. The Escape Trichotomy
Cantor set
Sierpinski curve
Cantor set of circles

3. Singular Perturbations of Complex Polynomials
2. Structures in the dynamical plane
All are Sierpinski curve Julia sets

4. Singular Perturbations of Complex Polynomials
3. Structures in the parameter planes

5. These lectures will deal with the dynamics
of the family of complex maps
where c is the center of a hyperbolic
component of the Multibrot set

6. These lectures will deal with the dynamics
of the family of complex maps
where c is the center of a hyperbolic
component of the Multibrot set
But for simplicity, we’ll concentrate for the
most part on the easier family

7. Why the interest in these maps?

8. Why the interest in these maps?
First, these are singular perturbations of

9. Why the interest in these maps?
First, these are singular perturbations of
When
but when , the dynamical behavior “explodes.”
we completely understand the dynamics of

10. Why the interest in these maps?
First, these are singular perturbations of
Second, how do you solve the equation ?

11. Why the interest in these maps?
First, these are singular perturbations of
Second, how do you solve the equation ?
You use Newton’s method (of course!):

12. Why the interest in these maps?
First, these are singular perturbations of
Second, how do you solve the equation ?
You use Newton’s method (of course!):
Iterate:

13. Why the interest in these maps?
First, these are singular perturbations of
Second, how do you solve the equation ?
You use Newton’s method (of course!):
Iterate:
a singular perturbation of z/2

14. Why the interest in these maps?
First, these are singular perturbations of
Second, how do you solve the equation ?
You use Newton’s method (of course!):
Whenever the equation has a multiple root, the corresponding Newton’s method involves a
singular perturbation.

15. Why the interest in these maps?
First, these are singular perturbations of
Second, how do you solve the equation ?
Third, we are looking at maps on the boundary of the
set of rational maps of degree 2n a very interesting
topic of contemporary research.

16. Dynamics of the family of complex maps
The Escape Trichotomy
Dynamics of the family of complex maps
with:
Paul Blanchard
Matt Holzer
U. Hoomiforgot
Dan Look
Sebastian Marotta
Monica Moreno Rocha
Yakov Shapiro
David Uminsky

17. A rational map of degree 2n.
Dynamics of
complex and
A rational map of degree 2n.

18. Dynamics of complex and A rational map of degree 2n. The Julia set is:
The closure of the set of repelling periodic points;
The boundary of the escaping orbits;
The chaotic set.
The Fatou set is the complement of

19. When , the Julia set
is the unit circle

20. When , the Julia set
is the unit circle
Colored points have orbits
that escape to infinity:
Escape time:
red (fastest)
orange
yellow
green
blue
violet (slowest)

21. When , the Julia set
is the unit circle
Colored points have orbits
that escape to infinity:
Escape time:
red (fastest)
orange
yellow
green
blue
violet (slowest)

22. When , the Julia set
is the unit circle
Colored points have orbits
that escape to infinity:
Escape time:
red (fastest)
orange
yellow
green
blue
violet (slowest)

23. When , the Julia set
is the unit circle
Colored points have orbits
that escape to infinity:
Escape time:
red (fastest)
orange
yellow
green
blue
violet (slowest)

24. When , the Julia set
is the unit circle
Colored points have orbits
that escape to infinity:
Escape time:
red (fastest)
orange
yellow
green
blue
violet (slowest)

25. When , the Julia set
is the unit circle
Colored points have orbits
that escape to infinity:
Escape time:
red (fastest)
orange
yellow
green
blue
violet (slowest)

26. When , the Julia set
is the unit circle
Colored points have orbits
that escape to infinity:
Escape time:
red (fastest)
orange
yellow
green
blue
violet (slowest)

27. When , the Julia set
is the unit circle
Colored points have orbits
that escape to infinity:
Escape time:
red (fastest)
orange
yellow
green
blue
violet (slowest)

28. When , the Julia set
is the unit circle
Black points have orbits
that do not escape to infinity:

29. When , the Julia set
is the unit circle
Black points have orbits
that do not escape to infinity:

30. When , the Julia set
is the unit circle
Black points have orbits
that do not escape to infinity:

31. When , the Julia set
is the unit circle
Black points have orbits
that do not escape to infinity:

32. When , the Julia set
is the unit circle
Black points have orbits
that do not escape to infinity:

33. When , the Julia set
is the unit circle
Black points have orbits
that do not escape to infinity:

34. When , the Julia set
is the unit circle
Black points have orbits
that do not escape to infinity:

35. When , the Julia set
is the unit circle
Black points have orbits
that do not escape to infinity:

36. When , the Julia set
is the unit circle
Black points have orbits
that do not escape to infinity:

37. When , the Julia set
is the unit circle
Black points have orbits
that do not escape to infinity:

38. When , the Julia set
is the unit circle
Black points have orbits
that do not escape to infinity:

39. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

40. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

41. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

42. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

43. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

44. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

45. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

46. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

47. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

48. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

49. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

50. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

51. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

52. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

53. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

54. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

55. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

56. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

57. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

58. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

59. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

60. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

61. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

62. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

63. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

64. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

65. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

66. When , the Julia set
is the unit circle
The Julia set is the boundary
of the black & colored regions.

67. When , the Julia set
is the unit circle
But when , the
Julia set explodes

68. When , the Julia set
is the unit circle
But when , the
Julia set explodes
A Sierpinski curve

69. When , the Julia set
is the unit circle
But when , the
Julia set explodes
Another Sierpinski
curve

70. When , the Julia set
is the unit circle
But when , the
Julia set explodes
Also a Sierpinski
curve

71. The Julia set has 2n-fold symmetry since
Easy computations:
The Julia set has 2n-fold
symmetry since
where is a 2nth root of unity

72. Easy computations:
2n free critical points

73. Easy computations:
2n free critical points

74. Easy computations:
2n free critical points
Only 2 critical values

75. Easy computations:
2n free critical points
Only 2 critical values

76. Easy computations:
2n free critical points
Only 2 critical values

77. Easy computations:
2n free critical points
Only 2 critical values
But really only 1 free
critical orbit since
the map has 2n-fold
symmetry

78. Easy computations: B
is superattracting, so
have immediate basin B
mapped n-to-1 to itself.

79. Easy computations: B
is superattracting, so
have immediate basin B
mapped n-to-1 to itself. T
0 is a pole, so have
trap door T mapped
n-to-1 to B.

80. So any orbit that eventually
Easy computations: B
is superattracting, so
have immediate basin B
mapped n-to-1 to itself. T
0 is a pole, so have
trap door T mapped
n-to-1 to B.
So any orbit that eventually
enters B must do so by
passing through T.

81. There are three distinct ways the critical orbit can enter B:
The Escape Trichotomy
There are three distinct ways the critical orbit can enter B:

82. There are three distinct ways the critical orbit can enter B:
The Escape Trichotomy
There are three distinct ways the critical orbit can enter B: B
is a Cantor set

83. The Escape Trichotomy
There are three distinct ways the critical orbit can enter B: B
is a Cantor set T
is a Cantor set of
simple closed curves
(this case does not occur if n = 2)

84. The Escape Trichotomy
There are three distinct ways the critical orbit can enter B: B
is a Cantor set T
is a Cantor set of
simple closed curves
(this case does not occur if n = 2) T
is a Sierpinski curve

85. parameter plane
when n = 3
Case 1: B
is a Cantor set

86. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

87. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

88. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

89. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

90. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

91. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

92. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

93. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

94. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

95. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

96. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

97. parameter plane
when n = 3
J is a Cantor set B
is a Cantor set

98. B is a Cantor set Draw curves connecting the
two critical values to in B

99. B is a Cantor set The preimages are curves
passing through the critical
points and connecting c to

100. B is a Cantor set The preimages are curves
passing through the critical
points and connecting c to and to 0

101. Choose a large circle in B
is a Cantor set
Choose a large circle in B

102. and locate its two preimagesB
is a Cantor set
and locate its two preimages

103. Construct the regions I0, ... I2n-1B
is a Cantor set I1 I2 I0 I3 I5 I4
Construct the regions I0, ... I2n-1

104. B is a Cantor set I1 I2 I0 I3 I5 I4 Each of the regions I0, ... I2n-1
is mapped 1-1 over the union
of the Ij

105. B is a Cantor set I1 I2 I0 I3 I5 I4 Each of the regions I0, ... I2n-1
is mapped 1-1 over the union
of the Ij

106. B is a Cantor set I1 I2 I0 I3 I5 I4 Each of the regions I0, ... I2n-1
is mapped 1-1 over the union
of the Ij

107. B is a Cantor set I1 I2 I0 I3 I5 I4 Each of the regions I0, ... I2n-1
is mapped 1-1 over the union
of the Ij

108. B is a Cantor set I1 I2 I0 I3 I5 I4 Each of the regions I0, ... I2n-1
is mapped 1-1 over the union
of the Ij
is a Cantor set

109. Case 2: the critical values lie in T, not B
parameter plane
when n = 3

110. lies in the McMullen domain
parameter plane
when n = 3 T
lies in the McMullen domain

111. lies in the McMullen domain
parameter plane
when n = 3
J is a Cantor set of
simple closed curves T
lies in the McMullen domain
Remark: There is no McMullen domain in the case n = 2.

112. lies in the McMullen domain
parameter plane
when n = 3
J is a Cantor set of
simple closed curves T
lies in the McMullen domain

113. lies in the McMullen domain
parameter plane
when n = 3
J is a Cantor set of
simple closed curves T
lies in the McMullen domain

114. lies in the McMullen domain
parameter plane
when n = 3
J is a Cantor set of
simple closed curves T
lies in the McMullen domain

115. lies in the McMullen domain
parameter plane
when n = 3
J is a Cantor set of
simple closed curves T
lies in the McMullen domain

116. lies in the McMullen domain
parameter plane
when n = 3
J is a Cantor set of
simple closed curves T
lies in the McMullen domain

117. lies in the McMullen domain
parameter plane
when n = 3
J is a Cantor set of
simple closed curves T
lies in the McMullen domain

118. J is a Cantor set of “circles”v c
Why is the preimage of T an annulus?

119. Could it be that each critical point lies in a disjoint preimage of T?

120. Could it be that each critical point lies in a disjoint preimage of T?
No. The map would then be 4n to 1 on the preimage of T.

121. By 2n-fold symmetry, there can then be only
Could it be that each critical point lies in a disjoint preimage of T?
No. The map would then be 4n to 1 on the preimage of T.
By 2n-fold symmetry, there can then be only
one preimage of T

122. Riemann-Hurwitz formula:
F: D R holomorphic branched covering
n(D), n(R) = number of boundary components of D, R

123. Riemann-Hurwitz formula:
F: D R holomorphic branched covering
n(D), n(R) = number of boundary components of D, R
then:
n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) *

124. In our case, R is the trap door and D is the
Riemann-Hurwitz formula:
F: D R holomorphic branched covering
n(D), n(R) = number of boundary components of D, R
then:
n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) *
In our case, R is the trap door and D is the
preimage of T

125. Riemann-Hurwitz formula:
F: D R holomorphic branched covering
n(D), n(R) = number of boundary components of D, R
then:
n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) *
We have deg F = 2n on D, so
n(D) - 2 = ( 2n ) (n(R) - 2) + (# of critical points in D) *

126. and R = T is a disk, so n(R) = 1
Riemann-Hurwitz formula:
F: D R holomorphic branched covering
n(D), n(R) = number of boundary components of D, R
then:
n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) *
and R = T is a disk, so n(R) = 1
n(D) - 2 = ( 2n ) (1 - 2) + (# of critical points in D) *

127. and there are 2n critical points in D
Riemann-Hurwitz formula:
F: D R holomorphic branched covering
n(D), n(R) = number of boundary components of D, R
then:
n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) *
and there are 2n critical points in D
n(D) - 2 = ( 2n ) (1 - 2) ( 2n ) = 0 *

128. and there are 2n critical points in D
Riemann-Hurwitz formula:
F: D R holomorphic branched covering
n(D), n(R) = number of boundary components of D, R
then:
n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) *
and there are 2n critical points in D
n(D) - 2 = ( 2n ) (1 - 2) ( 2n ) = 0 *
so n(D) = 2 and the preimage of T is an annulus.

129. J is a Cantor set of “circles”v c
So the preimage of T is an annulus.

130. J is a Cantor set of “circles”B T
B and T are mapped n-to-1 to B

131. J is a Cantor set of “circles”B T
The white annulus is mapped 2n-to-1 to T

132. J is a Cantor set of “circles”B T
So all that’s left are the blue annuli, and
each are mapped n-to-1 to the union of
the blue and white annuli.

133. J is a Cantor set of “circles”B T
So there are sub-annuli in the blue annuli
that are mapped onto the white annulus.

134. J is a Cantor set of “circles”

135. J is a Cantor set of “circles”

136. J is a Cantor set of “circles”
then all other preimages of F-1(T) contain no critical
points, and F is an n - to -1 covering on each, so
the remaining preimages of T are all annuli.

137. J is a Cantor set of “circles”
These annuli fill out the Fatou set; removing
all of them leaves us with a Cantor set of
simple closed curves (McMullen)

138. Curiously, this cannot happen when n = 2.

139. Reason: Modulus of an annulus1 r A
Any annulus in the plane is conformally isomorphic to a
unique round annulus of the form

140. Reason: Modulus of an annulus1 r A
Any annulus in the plane is conformally isomorphic to a
unique round annulus of the form
Then the modulus of A is

141. Reason: Modulus of an annulus1 r A
Any annulus in the plane is conformally isomorphic to a
unique round annulus of the form
Then the modulus of A is
And if F : A A2 is an analytic n-to-1 covering
map, then mod (A2) = n mod (A1)

142. This cannot happen when n = 2
Aout
Amid
Ain
2-to-1,

143. This cannot happen when n = 2
Aout
Amid
Ain
2-to-1, so the modulus of
is 1/2 the modulus of

144. This cannot happen when n = 2
Aout
Amid
Ain
2-to-1, so the modulus of
is 1/2 the modulus of
, same with

145. This cannot happen when n = 2
Aout
Amid
Ain
But then mod ( ) + mod ( ) = mod ( )
so there is no room for the middle annulus

146. Another reason this does not happen:
The critical values are
When n > 2 we have

147. Another reason this does not happen:
The critical values are
When n > 2 we have
so the critical value lies in T when is small.

148. Another reason this does not happen:
The critical values are
But when n = 2 we have

149. Another reason this does not happen:
The critical values are
But when n = 2 we have
So, as , and 1/4 is nowhere
near the basin of when is small

150. Parameter planes
n = 2
n = 3
No McMullen domain
McMullen domain

151. Parameter planes
n = 2
n = 3
No McMullen domain
McMullen domain

152. n = 2
n = 3
No McMullen domain
McMullen domain

153. n = 2
n = 3
No McMullen domain
McMullen domain

154. There is a lot of structure around the McMullen domain when n > 2
but a very different structure when n = 2.

155. Case 3: the critical orbit eventually lands in the trap door
is a “Sierpinski curve.”

156. Sierpinski Curve A Sierpinski curve is any planar
set that is homeomorphic to the Sierpinski carpet fractal.
The Sierpinski Carpet

157. Topological Characterization
Any planar set that is:
1. compact
2. connected
3. locally connected
4. nowhere dense
5. any two complementary
domains are bounded by
simple closed curves
that are pairwise disjoint
is a Sierpinski curve.
The Sierpinski Carpet

158. A Sierpinski curve is a universal plane continuum:
More importantly....
A Sierpinski curve is a universal plane continuum:
Any planar, one-dimensional, compact, connected set can be homeomorphically embedded in a Sierpinski curve.
For example....

159. The topologist’s sine curve
can be embedded inside

160. The topologist’s sine curve
can be embedded inside

161. The topologist’s sine curve
can be embedded inside
skip Knaster

162. The Knaster continuum
can be embedded inside

163. Start with the Cantor middle-thirds set
Knaster continuum
A well known example of an indecomposable continuum
Start with the Cantor middle-thirds set

164. Knaster continuum
Connect symmetric points about 1/2 with semicircles

165. Knaster continuum
Do the same below about 5/6

166. Knaster continuum
And continue....

167. Knaster continuum

168. Properties of K: There is one curve that passes through all the
endpoints of the Cantor
set.

169. Properties of K: There is one curve that passes through all the
endpoints of the Cantor
set.
It accumulates everywhere on
itself and on K.

170. Properties of K: There is one curve that passes through all the
endpoints of the Cantor
set.
It accumulates everywhere on
itself and on K.
And is the only piece
of K that is accessible
from the outside.

171. Properties of K: There is one curve that passes through all the
endpoints of the Cantor
set.
It accumulates everywhere on
itself and on K.
And is the only piece
of K that is accessible
from the outside.
But there are infinitely many other curves
in K, each of which is dense in K.

172. The Knaster continuum
can be embedded inside

173. Note the Cantor middle-thirds set along the middle horizontal line
The Knaster continuum
can be embedded inside
Note the Cantor middle-thirds set
along the middle horizontal line

174. The Knaster continuum
can be embedded inside

175. The Knaster continuum
can be embedded inside

176. The Knaster continuum
can be embedded inside

177. The Knaster continuum
can be embedded inside

178. The Knaster continuum
can be embedded inside

179. The Knaster continuum
can be embedded inside

180. The Knaster continuum
can be embedded inside

181. The Knaster continuum
can be embedded inside

182. The Knaster continuum
can be embedded inside

183. The Knaster continuum
can be embedded inside

184. The Knaster continuum
can be embedded inside

185. The Knaster continuum
can be embedded inside

186. The Knaster continuum
can be embedded inside

187. The Knaster continuum
can be embedded inside

188. The Knaster continuum
can be embedded inside

189. The Knaster continuum
can be embedded inside

190. The Knaster continuum
can be embedded inside

191. The Knaster continuum
can be embedded inside

192. The Knaster continuum
can be embedded inside

193. The Knaster continuum
can be embedded inside

194. The Knaster continuum
can be embedded inside

195. The Knaster continuum
can be embedded inside

196. The Knaster continuum
can be embedded inside

197. The Knaster continuum
can be embedded inside

198. The Knaster continuum
can be embedded inside

199. The Knaster continuum
can be embedded inside

200. The Knaster continuum
can be embedded inside

201. The Knaster continuum
can be embedded inside

202. The Knaster continuum
can be embedded inside

203. The Knaster continuum
can be embedded inside

204. The Knaster continuum
can be embedded inside

205. The Knaster continuum
can be embedded inside

206. The Knaster continuum
can be embedded inside

207. Even this “curve”
can be embedded inside

208. can be embedded inside Even this “curve” Moreover, Sierpinski curves
occur all the time as Julia sets.

209. lies in a Sierpinski hole
Case 3: the critical orbit eventually lands in the trap door.
parameter plane
when n = 3 T
lies in a Sierpinski hole

210. lies in a Sierpinski hole
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

211. lies in a Sierpinski hole
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

212. lies in a Sierpinski hole
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

213. lies in a Sierpinski hole
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

214. lies in a Sierpinski hole
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

215. lies in a Sierpinski hole
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

216. lies in a Sierpinski hole
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

217. lies in a Sierpinski hole
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

218. lies in a Sierpinski hole
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

219. lies in a Sierpinski hole
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

220. lies in a Sierpinski hole
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

221. To show that
is homeomorphic to

222. Need to show:
compact
connected
nowhere dense
locally connected
bounded by disjoint
s.c.c.’s

223. Need to show:
compact
connected
nowhere dense
locally connected
bounded by disjoint
s.c.c.’s
Fatou set is the union of the preimages of B; all
disjoint, open disks.

224. Need to show:
compact
connected
nowhere dense
locally connected
bounded by disjoint
s.c.c.’s
Fatou set is the union of the preimages of B; all
disjoint, open disks.

225. Need to show:
compact
connected
nowhere dense
locally connected
bounded by disjoint
s.c.c.’s
If J contains an open
set, then J = C.

226. Need to show:
compact
connected
nowhere dense
locally connected
bounded by disjoint
s.c.c.’s
If J contains an open
set, then J = C.

227. Need to show:
compact
connected
nowhere dense
locally connected
bounded by disjoint
s.c.c.’s
No recurrent critical orbits, no parabolic points., and F
is hyperbolic on J....

228. Need to show:
compact
connected
nowhere dense
locally connected
bounded by disjoint
s.c.c.’s
No recurrent critical orbits, no parabolic points., and F
is hyperbolic on J....

229. Need to show:
compact
connected
nowhere dense
locally connected
bounded by disjoint
s.c.c.’s
J locally connected, so the
boundaries are locally connected. Need to show they are s.c.c.’s. Can only meet at (preimages of) critical points, hence disjoint.

230. Need to show:
compact
connected
nowhere dense
locally connected
bounded by disjoint
s.c.c.’s
So J is a Sierpinski curve.

231. Have an exact count of the number of Sierpinski holes:
Theorem (Roesch): Given n, there are exactly
(n-1)(2n) Sierpinski holes with escape time k.
(k-3)

232. Have an exact count of the number of Sierpinski holes:
Theorem (Roesch): Given n, there are exactly
(n-1)(2n) Sierpinski holes with escape time k.
(k-3)
Reason: The equation
reduces to a polynomial of degree (n-1)(2n)(k-3) ,
and it can be shown that all the roots of this
polynomial are distinct. (You can put a Böttcher
coordinate on each Sierpinski hole).

233. Have an exact count of the number of Sierpinski holes:
Theorem (Roesch): Given n, there are exactly
(n-1)(2n) Sierpinski holes with escape time k.
(k-3)
Reason: The equation
reduces to a polynomial of degree (n-1)(2n)(k-3) ,
and it can be shown that all the roots of this
polynomial are distinct. (You can put a Böttcher
coordinate on each Sierpinski hole). So we have
exactly that many “centers” of Sierpinski holes,
i.e., parameters for which the critical points all
land on 0 and then

234. Have an exact count of the number of Sierpinski holes:
Theorem (Roesch): Given n, there are exactly
(n-1)(2n) Sierpinski holes with escape time k.
(k-3)
n = 3
escape time 3
2 Sierpinski holes
parameter plane
n = 3

235. Have an exact count of the number of Sierpinski holes:
Theorem (Roesch): Given n, there are exactly
(n-1)(2n) Sierpinski holes with escape time k.
(k-3)
n = 3
escape time 3
2 Sierpinski holes
parameter plane
n = 3

236. Have an exact count of the number of Sierpinski holes:
Theorem (Roesch): Given n, there are exactly
(n-1)(2n) Sierpinski holes with escape time k.
(k-3)
n = 3
escape time 4
12 Sierpinski holes
parameter plane
n = 3

237. Have an exact count of the number of Sierpinski holes:
Theorem (Roesch): Given n, there are exactly
(n-1)(2n) Sierpinski holes with escape time k.
(k-3)
n = 3
escape time 4
12 Sierpinski holes
parameter plane
n = 3

238. Have an exact count of the number of Sierpinski holes:
Theorem (Roesch): Given n, there are exactly
(n-1)(2n) Sierpinski holes with escape time k.
(k-3)
n = 4
escape time 3
3 Sierpinski holes
parameter plane
n = 4

239. Have an exact count of the number of Sierpinski holes:
Theorem (Roesch): Given n, there are exactly
(n-1)(2n) Sierpinski holes with escape time k.
(k-3)
n = 4
escape time 4
24 Sierpinski holes
parameter plane
n = 4

240. Have an exact count of the number of Sierpinski holes:
Theorem (Roesch): Given n, there are exactly
(n-1)(2n) Sierpinski holes with escape time k.
(k-3)
n = 4
escape time 12
402,653,184
Sierpinski holes
Sorry. I forgot to
indicate their locations.
parameter plane
n = 4

241. Given two Sierpinski curve Julia
sets, when do we know that the dynamics on them are
the same, i.e., the maps are conjugate on the Julia sets?
Main Question:
These sets are homeomorphic, but are
the dynamics on them the same?

242. Given two Sierpinski curve Julia
sets, when do we know that the dynamics on them are
the same, i.e., the maps are conjugate on the Julia sets?
Main Question:
We know when two maps with Sierpinski curve Julia sets
are conjugate on their Julia sets (next lecture), but:
Problem: What is the different dynamical behavior when
two such maps are not conjugate on their Julia sets. Clearly
a job for symbolic dynamics.....

243. Other ways that Sierpinski curve Julia sets arise:
Buried points in Cantor necklaces;
2. Main cardioids in buried Mandelbrot sets;
Cantor sets of circles around the McMullen domain;
4. Other families of rational maps;
5. Sierpinski gaskets
6. Major applications

244. 1. Cantor necklaces in the parameter plane
The necklace is the Cantor middle thirds
set with disks replacing removed intervals.
parameter plane
n = 4

245. 1. Cantor necklaces in the parameter plane
There is a Cantor necklace
along the negative real axis
parameter plane
n = 4

246. 1. Cantor necklaces in the parameter plane5 5 4 3 4
The open disks are Sierpinski holes
parameter plane
n = 4

247. 1. Cantor necklaces in the parameter plane5 5 4 3 4
The open disks are Sierpinski holes;
the buried points in the Cantor set
also correspond to Sierpinski curves;
parameter plane
n = 4

248. 1. Cantor necklaces in the parameter plane5 5 4 3 4
The open disks are Sierpinski holes;
the buried points in the Cantor set
also correspond to Sierpinski curves;
and all are dynamically different.
parameter plane
n = 4

249. 1. Cantor necklaces in the parameter plane
A similar Cantor necklace when n = 2
The parameter plane when n = 2

250. 1. Cantor necklaces in the parameter plane
A similar Cantor necklace when n = 2
The parameter plane when n = 2

251. 1. Cantor necklaces in the parameter plane6 6 5 4 5
A similar Cantor necklace when n = 2
The parameter plane when n = 2

252. 1. Cantor necklaces in the parameter plane
When the parameter is negative, the
critical points do not lie on the real
axis;
A Sierpinski curve from a buried
point in the Cantor necklace when n=2

253. 1. Cantor necklaces in the parameter plane
When the parameter is negative, the
critical points do not lie on the real
axis;
A Sierpinski curve from a buried
point in the Cantor necklace when n=2

254. 1. Cantor necklaces in the parameter plane
When the parameter is negative, the
critical points do not lie on the real
axis; they map to the imaginary axis;
A Sierpinski curve from a buried
point in the Cantor necklace when n=2

255. 1. Cantor necklaces in the parameter plane
When the parameter is negative, the
critical points do not lie on the real
axis; they map to the imaginary axis;
and then to the real axis, where the
orbit then remains forever.
A Sierpinski curve from a buried
point in the Cantor necklace when n=2

256. 1. Cantor necklaces in the parameter plane
When the parameter is negative, the
critical points do not lie on the real
axis; they map to the imaginary axis;
and then to the real axis, where the
orbit then remains forever. So the
critical orbits are non-recurrrent
J is locally connected.
A Sierpinski curve from a buried
point in the Cantor necklace when n=2

257. 1. Cantor necklaces in the parameter plane
The only Fatou components are B
and its preimages since the critical
orbits lie in a Cantor set which is
not in the Fatou set J is
compact and connected.
A Sierpinski curve from a buried
point in the Cantor necklace when n=2

258. 1. Cantor necklaces in the parameter plane
There is a Fatou component
J is nowhere dense
A Sierpinski curve from a buried
point in the Cantor necklace when n=2

259. 1. Cantor necklaces in the parameter plane
There is a Fatou component
J is nowhere dense, and same
arguments as before show that
the boundary consists of disjoint
simple closed curves
J is a Sierpinski curve.
A Sierpinski curve from a buried
point in the Cantor necklace when n=2

260. 1. Cantor necklaces in the parameter plane
The “endpoints” in the Cantor set
(parameters on the boundaries of
the Sierpinski holes) do not correspond
to Sierpinski curves.
A “hybrid” Sierpinski curve;
some boundary curves meet.

261. 1. Cantor necklaces in the parameter plane
The “endpoints” in the Cantor set
(parameters on the boundaries of
the Sierpinski holes) do not correspond
to Sierpinski curves. Certain
preimages of T touch each other at
(pre)-critical points.
A “hybrid” Sierpinski curve;
some boundary curves meet.

262. 1. Cantor necklaces in the parameter plane
There are lots of other Cantor
necklaces in the parameter planes.
parameter plane
n = 2

263. 1. Cantor necklaces in the parameter plane
There are lots of other Cantor
necklaces in the parameter planes.
parameter plane
n = 2

264. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
n = 4

265. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
n = 4

266. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
n = 4

267. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

268. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

269. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

270. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

271. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

272. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

273. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

274. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

275. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

276. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

277. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

278. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

279. If lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
A Sierpinski curve, but very different
dynamically from the earlier ones.
n = 4

280. If n > 2, there are uncountably many simple
closed curves surrounding the McMullen domain;
all these parameters are Sierpinski curves.
n = 3

281. If n > 2, there are uncountably many simple
closed curves surrounding the McMullen domain;
all these parameters are Sierpinski curves.
n = 3

282. If n > 2, there are uncountably many simple
closed curves surrounding the McMullen domain;
all these parameters are Sierpinski curves.
All parameters on these curves
correspond to Sierpinski curves.
n = 3

283. If n > 2, there are uncountably many simple
closed curves surrounding the McMullen domain;
all these parameters are Sierpinski curves.
And all non-symmetrically located
parameters on these curves have
very different dynamics on J.
n = 3

284. If n > 2, there are also countably many different simple closed
curves accumulating on the McMullen domain. Each
alternately passes through centers of Sierpinski
holes and centers of baby Mandelbrot sets.
2 centers
n = 3

285. If n > 2, there are also countably many different simple closed
curves accumulating on the McMullen domain. Each
alternately passes through centers of Sierpinski
holes and centers of baby Mandelbrot sets.
4 centers
n = 3

286. If n > 2, there are also countably many different simple closed
curves accumulating on the McMullen domain. Each
alternately passes through centers of Sierpinski
holes and centers of baby Mandelbrot sets.
10 centers
n = 3

287. If n > 2, there are also countably many different simple closed
curves accumulating on the McMullen domain. Each
alternately passes through centers of Sierpinski
holes and centers of baby Mandelbrot sets.
28 centers
n = 3

288. If n > 2, there are also countably many different simple closed
curves accumulating on the McMullen domain. Each
alternately passes through centers of Sierpinski
holes and centers of baby Mandelbrot sets.
82 centers
n = 3

289. If n > 2, there are also countably many different simple closed
curves accumulating on the McMullen domain. Each
alternately passes through centers of Sierpinski
holes and centers of baby Mandelbrot sets.
passes through
1,594,324 centers.
n = 3

290. If n > 2, there are also countably many different simple closed
curves accumulating on the McMullen domain. Each
alternately passes through centers of Sierpinski
holes and centers of baby Mandelbrot sets.
As before, all non-
symmetrically located
centers have
different dynamics.
n = 3

291. Consider the family of maps
where c is the center of a hyperbolic component
of the Mandelbrot set.

292. Consider the family of maps
where c is the center of a hyperbolic component
of the Mandelbrot set.

293. Consider the family of maps
where c is the center of a hyperbolic component
of the Mandelbrot set.

294. , the Julia set again expodes.
When
, the Julia set again expodes.

295. , the Julia set again expodes.
When
, the Julia set again expodes.

296. , the Julia set again expodes.
When
, the Julia set again expodes.

297. , the Julia set again expodes.
When
, the Julia set again expodes.

298. , the Julia set again expodes.
When
, the Julia set again expodes.
A doubly-inverted
Douady rabbit.

299. If you chop off the “ears” of each internal rabbit in each component
of the original Julia set, then what’s left is another Sierpinski curve
(provided that both of the critical orbits eventually escape).

300. The (generalized) Sierpinski gasket also occurs
(although rarely) as a Julia set in these families

301. Choose at the tip of the parameter plane on the negative real axis
Parameter plane for
n=2, d=1

302. This Julia set is homeomorphic to the gasket

303. If is a Misiurewicz parameter on the boundary of the
connectedness locus, then is a “generalized” gasket.
Parameter plane for
n=2, d=1

304. If is a Misiurewicz parameter on the boundary of the
connectedness locus, then is a “generalized” gasket.
Parameter plane for
n=2, d=1

305. If is a Misiurewicz parameter on the boundary of the
connectedness locus, then is a “generalized” gasket.
Parameter plane for
n=2, d=1

306. None of these other Julia sets are homeomorphic
to the Sierpinski gasket, however.

307. None of these other Julia sets are homeomorphic
to the Sierpinski gasket, however.

308. None of these other Julia sets are homeomorphic
to the Sierpinski gasket, however. T
The “triangle” T touches the
inner triangle twice, not once.

309. For other values of n and d, the generalized
Sierpinski gaskets are not triangular. T
n = d = 2
A “square” gasket

310. For other values of n and d, the generalized
Sierpinski gaskets are not triangular.
n = 3, d = 2
A “pentagonal” gasket

311. Here’s the parameter plane when n = 2:

312. and this object appears everywhere.....
Rotate it by 90 degrees:
and this object appears everywhere.....