Dynamics of the family of complex maps

1
Dynamics of Singularly Perturbed Rational Maps
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
Dynamics of Singularly Perturbed Rational Maps
1. The Escape Trichotomy
Cantor set
Sierpinski curve
Cantor set of circles
3
Dynamics of Singularly Perturbed Rational Maps
2. Classification of escape time Sierpinski
curve Julia sets
All are Sierpinski curve Julia sets, but with
very different dynamics
4
Dynamics of Singularly Perturbed Rational Maps
3. Julia sets converging to the unit disk
? -.0001
? -.000001
? -.01
5
Dynamics of Singularly Perturbed Rational Maps
4. Cantor necklaces and webs
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
7
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, well concentrate for the
most part on the easier family
8
Why the interest in these maps?

9
Why the interest in these maps?
First, these are singular perturbations of
.
10
Why the interest in these maps?
First, these are singular perturbations of
.
we completely understand the dynamics of
When but when , the dynamical behavior
explodes.
11
Why the interest in these maps?
First, these are singular perturbations of
.
Second, how do you solve the equation
?
12
Why the interest in these maps?
First, these are singular perturbations of
.
Second, how do you solve the equation
?
You use Newtons method (of course!)
13
Why the interest in these maps?
First, these are singular perturbations of
.
Second, how do you solve the equation
?
You use Newtons method (of course!)
Iterate
14
Why the interest in these maps?
First, these are singular perturbations of
.
Second, how do you solve the equation
?
You use Newtons method (of course!)
Iterate
a singular perturbation of z/2
15
Why the interest in these maps?
First, these are singular perturbations of
.
Second, how do you solve the equation
?
You use Newtons method (of course!)
Whenever the equation has a
multiple root, the corresponding Newtons method
involves a singular perturbation.
16
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.
17
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
But when , the Julia set explodes
When , the Julia set is the unit circle
68
But when , the Julia set explodes
When , the Julia set is the unit circle
A Sierpinski curve
69
But when , the Julia set explodes
When , the Julia set is the unit circle
Another Sierpinski curve
70
But when , the Julia set explodes
When , the Julia set is the unit circle
Also a Sierpinski curve
71
Easy computations
J(F?) has 2n-fold symmetry
72
Easy computations
J(F?) has 2n-fold symmetry
2n free critical points
73
Easy computations
J(F?) has 2n-fold symmetry
2n free critical points
74
Easy computations
J(F?) has 2n-fold symmetry
2n free critical points
Only 2 critical values
75
Easy computations
J(F?) has 2n-fold symmetry
2n free critical points
Only 2 critical values
76
Easy computations
J(F?) has 2n-fold symmetry
2n free critical points
Only 2 critical values
77
Easy computations
J(F?) has 2n-fold symmetry
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
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
Part 1. 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
82
1. The Escape Trichotomy
There are three distinct ways the critical
orbit can enter B
83
1. The Escape Trichotomy
There are three distinct ways the critical
orbit can enter B
B
is a Cantor set
84
1. 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)
85
1. 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
86
parameter plane when n 3
B
Case 1
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
parameter plane when n 3
J is a Cantor set
B
is a Cantor set
99
B
is a Cantor set
v1
v2
Draw symmetric curves connecting the two
critical values to in B
100
B
is a Cantor set
v1
v2
The preimages are curves passing through the
critical points and connecting c to
101
B
is a Cantor set
The preimages are curves passing through the
critical points and connecting c to and
to 0
102
B
is a Cantor set
Choose a large circle in B
103
B
is a Cantor set
and locate its two preimages
104
B
is a Cantor set
I1
I0
I2
I3
I5
I4
Construct the regions I0, ... I2n-1
105
B
is a Cantor set
I1
I0
I2
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
I0
I2
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
I0
I2
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
I0
I2
I3
I5
I4
Each of the regions I0, ... I2n-1 is mapped 1-1
over the union of the Ij
109
B
is a Cantor set
I1
I0
I2
I3
I5
I4
Each of the regions I0, ... I2n-1 is mapped 1-1
over the union of the Ij
is a Cantor set
110
Case 2 the critical values lie in T, not B
parameter plane when n 3
111
parameter plane when n 3
T
lies in the McMullen domain
112
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.
113
parameter plane when n 3
J is a Cantor set of simple closed curves
T
lies in the McMullen domain
114
parameter plane when n 3
J is a Cantor set of simple closed curves
T
lies in the McMullen domain
115
parameter plane when n 3
J is a Cantor set of simple closed curves
T
lies in the McMullen domain
116
parameter plane when n 3
J is a Cantor set of simple closed curves
T
lies in the McMullen domain
117
parameter plane when n 3
J is a Cantor set of simple closed curves
T
lies in the McMullen domain
118
parameter plane when n 3
J is a Cantor set of simple closed curves
T
lies in the McMullen domain
119
T
J is a Cantor set of circles
v
c
Why is the preimage of T an annulus?
120
Could it be that each critical point lies in a
disjoint preimage of T?
121
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.
122
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
123
Riemann-Hurwitz formula
F D R holomorphic branched covering
n(D), n(R) number of boundary components of D,
R
124
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)

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)

In our case, R is the trap door and D ??? is
the preimage of T
126
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)

127
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)

128
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

129
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.
130
T
J is a Cantor set of circles
v
c
So the preimage of T is an annulus.
131
T
J is a Cantor set of circles
B
T
B and T are mapped n-to-1 to B
132
T
J is a Cantor set of circles
B
T
The white annulus is mapped 2n-to-1 to T
133
T
J is a Cantor set of circles
B
T
So all thats left are the blue annuli, and each
are mapped n-to-1 to the union of the blue and
white annuli.
134
T
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.
135
T
J is a Cantor set of circles
2n to 1
136
T
J is a Cantor set of circles
n to 1
n to 1
2n to 1
137
T
J is a Cantor set of circles
n to 1
n to 1
2n to 1
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.
138
T
J is a Cantor set of circles
n to 1
n to 1
2n to 1
These annuli fill out the Fatou set
removing all of them leaves us with a Cantor set
of simple closed curves (McMullen)
139
Curiously, this cannot happen when n 2. One
reason involves the moduli of the annuli in the
preimages of T.
140
Case n 2
A
B
T
A is mapped 4-to-1 to T
141
Case n 2
A1
A
B
A0
T
A0 and A1 are mapped 2-to-1 to A ? A0 ?A1
142
Case n 2
A1
A
B
A0
T
So mod(A0) mod( A ? A0 ?A1 ) and mod(A1)
mod( A ? A0 ?A1 )
143
Case n 2
A1
A
B
A0
T
So mod(A0) mod( A ? A0 ?A1 ) and mod(A1)
mod( A ? A0 ?A1 ) so there is no room in the
middle for A
144
Another reason this does not happen
The critical values are
When n gt 2 we have
145
Another reason this does not happen
The critical values are
When n gt 2 we have
so the critical values lie in T when is
small.
146
Another reason this does not happen
The critical values are
But when n 2 we have
147
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.
148
Parameter planes
n 2
n 3
No McMullen domain
McMullen domain
149
Parameter planes
n 2
n 3
No McMullen domain
McMullen domain
150
n 2
n 3
No McMullen domain
McMullen domain
151
n 2
n 3
No McMullen domain
McMullen domain
152
There is a lot of structure around the McMullen
domain when n gt 2
but a very different structure when n 2.
153
Case 3 the critical orbit eventually lands in
the trap door
is a Sierpinski curve.
154
Sierpinski Curve
A Sierpinski curve is any planar set that is
homeomorphic to the Sierpinski carpet fractal.
The Sierpinski Carpet
155
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
156
Case 3 the critical orbit eventually lands in
the trap door.
parameter plane when n 3
T
lies in a Sierpinski hole
157
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
158
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
159
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
160
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
161
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
162
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
163
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
164
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
165
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
166
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
167
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
168
Have an exact count of the number of Sierpinski
holes
Theorem (Roesch) Given n, there are exactly
(n-1)(2n)k-3 Sierpinski holes with escape time
k.
169
Have an exact count of the number of Sierpinski
holes
Theorem (Roesch) Given n, there are exactly
(n-1)(2n)k-3 Sierpinski holes with escape time
k.
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).
170
Have an exact count of the number of Sierpinski
holes
Theorem (Roesch) Given n, there are exactly
(n-1)(2n)k-3 Sierpinski holes with escape time
k.
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 .
171
Have an exact count of the number of Sierpinski
holes
Theorem (Roesch) Given n, there are exactly
(n-1)(2n)k-3 Sierpinski holes with escape time
k.
n 3 escape time 3 2 Sierpinski holes
parameter plane n 3
172
Have an exact count of the number of Sierpinski
holes
Theorem (Roesch) Given n, there are exactly
(n-1)(2n)k-3 Sierpinski holes with escape time
k.
n 3 escape time 3 2 Sierpinski holes
parameter plane n 3
173
Have an exact count of the number of Sierpinski
holes
Theorem (Roesch) Given n, there are exactly
(n-1)(2n)k-3 Sierpinski holes with escape time
k.
n 3 escape time 4 12 Sierpinski holes
parameter plane n 3
174
Have an exact count of the number of Sierpinski
holes
Theorem (Roesch) Given n, there are exactly
(n-1)(2n)k-3 Sierpinski holes with escape time
k.
n 3 escape time 4 12 Sierpinski holes
parameter plane n 3
175
Have an exact count of the number of Sierpinski
holes
Theorem (Roesch) Given n, there are exactly
(n-1)(2n)k-3 Sierpinski holes with escape time
k.
n 4 escape time 3 3 Sierpinski holes
parameter plane n 4
176
Have an exact count of the number of Sierpinski
holes
Theorem (Roesch) Given n, there are exactly
(n-1)(2n)k-3 Sierpinski holes with escape time
k.
n 4 escape time 4 24 Sierpinski holes
parameter plane n 4
177
Have an exact count of the number of Sierpinski
holes
Theorem (Roesch) Given n, there are exactly
(n-1)(2n)k-3 Sierpinski holes with escape time
k.
n 4 escape time 12 402,653,184 Sierpinski
holes
Sorry. I forgot to indicate their locations.
parameter plane n 4
178
Main Question
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?
These sets are homeomorphic, but are the
dynamics on them the same?
179
Part 2. Dynamic Classification of Escape
Time Sierpinski Curve Julia Sets
J ( F? )
J ( F? )
When do F? and F? have the same (conjugate)
dynamics?
180
1 If and are drawn from the same
Sierpinski hole, then the corresponding maps
have the same dynamics, i.e., they are
topologically conjugate on their Julia sets.
parameter plane n 4
181
1 If and are drawn from the same
Sierpinski hole, then the corresponding maps
have the same dynamics, i.e., they are
topologically conjugate on their Julia sets.
So all these parameters have the same dynamics on
their Julia sets.
parameter plane n 4
182
1 If and are drawn from the same
Sierpinski hole, then the corresponding maps
have the same dynamics, i.e., they are
topologically conjugate on their Julia sets.
This uses quasiconformal surgery techniques
parameter plane n 4
183
1 If and are drawn from the same
Sierpinski hole, then the corresponding maps
have the same dynamics, i.e., they are
topologically conjugate on their Julia sets.
2 If these parameters come from Sierpinski
holes with different escape times, then the
maps cannot be conjugate.
184
Two Sierpinski curve Julia sets, so they are
homeomorphic.
185
escape time 3
escape time 4
So these maps cannot be topologically conjugate.
186
is the only invariant boundary of an
escape component, so must be preserved by any
conjugacy.
187
is the only preimage of , so this curve
must also be preserved by a conjugacy.
188
If a boundary component is mapped to after k
iterations, its image under the conjugacy must
also have this property, and so forth.....
189
3-1
1-1
2-1
c
The curves around c are special they are the
only other ones in J mapped 2-1 onto their images.
190
3-1
3-1
1-1
1-1
1-1
2-1
2-1
c
This bounding region takes 3 iterates to land on
the boundary of B.
But this bounding region takes 4 iterates to
land, so these maps are not conjugate.
191
3 What if two maps lie in different Sierpinski
holes that have the same escape time?
For this it suffices to consider the centers of
the Sierpinski holes i.e., parameter values for
which for some k ? 3.
192
3 What if two maps lie in different Sierpinski
holes that have the same escape time?
For this it suffices to consider the centers of
the Sierpinski holes i.e., parameter values for
which for some k ? 3.
Two such centers of Sierpinski holes are
critically finite maps, so by Thurstons
Theorem, if they are topologically conjugate in
the plane, they can be conjugated by a Mobius
transformation (in the orientation preserving
case).
193
3 What if two maps lie in different Sierpinski
holes that have the same escape time?
For this it suffices to consider the centers of
the Sierpinski holes i.e., parameter values for
which for some k ? 3.
Two such centers of Sierpinski holes are
critically finite maps, so by Thurstons
Theorem, if they are topologically conjugate in
the plane, they can be conjugated by a Mobius
transformation (in the orientation preserving
case).
Since 8 ? 8 and 0 ? 0 under the
conjugacy, the Mobius conjugacy must be of the
form z ? ?z .
194
If we have a conjugacy
then
195
If we have a conjugacy
then
Comparing coefficients
196
If we have a conjugacy
then
Comparing coefficients
197
If we have a conjugacy
then
Comparing coefficients
Easy check --- for the orientation reversing case
is conjugate to via
198
Theorem. If and are centers of
Sierpinski holes, then iff
or where is a
primitive root of unity then any
two parameters drawn from these holes have the
same dynamics.
n 3
Only and are conjugate centers since
,
,
,
,
,
n 4
Only
are conjugate centers where .
199
n 3, escape time 4, 12 Sierpinski holes, but
only six conjugacy classes
conjugate centers
,
200
n 3, escape time 4, 12 Sierpinski holes, but
only six conjugacy classes
conjugate centers
,
201
n 3, escape time 4, 12 Sierpinski holes, but
only six conjugacy classes
conjugate centers
,
202
n 3, escape time 4, 12 Sierpinski holes, but
only six conjugacy classes
conjugate centers
,
203
n 4, escape time 4, 24 Sierpinski holes, but
only five conjugacy classes
conjugate centers
,
,
,
,
,
where
204
n 4, escape time 4, 24 Sierpinski holes, but
only five conjugacy classes
conjugate centers
,
,
,
,
,
where
205
n 4, escape time 4, 24 Sierpinski holes, but
only five conjugacy classes
conjugate centers
,
,
,
,
,
where
206
n 4, escape time 4, 24 Sierpinski holes, but
only five conjugacy classes
conjugate centers
,
,
,
,
,
where
207
n 4, escape time 4, 24 Sierpinski holes, but
only five conjugacy classes
conjugate centers
,
,
,
,
,
where
208
k-3
Theorem For any n there are exactly (n-1)
(2n) Sierpinski holes with escape time k.
The number of distinct conjugacy classes is given
by
k-3
a. (2n) when n is odd
k-3
k-4
b. (2n) /2 2 when n is even.
209
For n odd, there are no Sierpinski holes along
the real axis, so there are exactly n - 1
conjugate Sierpinski holes.
n 3
n 5
210
For n even, there is a Cantor necklace along
the negative axis, so there are some real
Sierpinski holes,
n 4
211
For n even, there is a Cantor necklace along
the negative axis, so there are some real
Sierpinski holes,
n 4
212
For n even, there is a Cantor necklace along
the negative axis, so there are some real
Sierpinski holes,
n 4
magnification
M
213
For n even, there is a Cantor necklace along
the negative axis, so there are some real
Sierpinski holes,
n 4
magnification
5
5
M
3
4
4
214
For n even, there is a Cantor necklace along
the negative axis, so we can count the number of
real Sierpinski holes, and there are exactly n
- 1 conjugate holes in this case
n 4
magnification
5
5
M
3
4
4
215
For n even, there are also 2(n - 1) complex
Sierpinski holes that have conjugate dynamics
n 4
magnification
M
216
n 4 402,653,184 Sierpinski holes with escape
time 12 67,108,832 distinct conjugacy classes.
Sorry. I again forgot to indicate their
locations.
217
n 4 402,653,184 Sierpinski holes with escape
time 12 67,108,832 distinct conjugacy classes.
Problem Describe the dynamics on these
different conjugacy classes.
218
Part 3 Julia sets converging to the unit disk
With Toni Garijo, Mark Morabito, and Robert T.
Kozma
219
Part 3 Julia sets converging to the unit disk
With Toni Garijo, Mark Morabito, and Robert T.
Kozma
n 2 When , the Julia set of
is the unit circle. But, as , the
Julia set of converges to the closed unit
disk
220
Part 3 Julia sets converging to the unit disk
With Toni Garijo, Mark Morabito, and Robert T.
Kozma
n 2 When , the Julia set of
is the unit circle. But, as , the
Julia set of converges to the closed unit
disk
n gt 2 J is always a Cantor set of circles
when is small.
221
Part 3 Julia sets converging to the unit disk
With Toni Garijo, Mark Morabito, and Robert T.
Kozma
n 2 When , the Julia set of
is the unit circle. But, as , the
Julia set of converges to the closed unit
disk
n gt 2 J is always a Cantor set of circles
when is small.
Moreover, there is a ? gt 0 such that there is
always a round annulus of thickness ?
between two of these circles in the Fatou set.
So J does not converge to the unit disk when
n gt 2.
222
Theorem When n 2, the Julia sets converge to
the unit disk as
n 2
223
Sketch of the proof
Suppose the Julia sets do not converge to the
unit disk D as
224
Sketch of the proof
Suppose the Julia sets do not converge to the
unit disk D as
Then there exists and a sequence
such that, for each i, there is a point
such that lies in the
Fatou set.
225
Sketch of the proof
Suppose the Julia sets do not converge to the
unit disk D as
Then there exists and a sequence
such that, for each i, there is a point
such that lies in the
Fatou set.
226
Sketch of the proof
Suppose the Julia sets do not converge to the
unit disk D as
Then there exists and a sequence
such that, for each i, there is a point
such that lies in the
Fatou set.
227
Sketch of the proof
Suppose the Julia sets do not converge to the
unit disk D as
Then there exists and a sequence
such that, for each i, there is a point
such that lies in the
Fatou set.
228
Sketch of the proof
Suppose the Julia sets do not converge to the
unit disk D as
Then there exists and a sequence
such that, for each i, there is a point
such that lies in the
Fatou set.
229
Sketch of the proof
Suppose the Julia sets do not converge to the
unit disk D as
Then there exists and a sequence
such that, for each i, there is a point
such that lies in the
Fatou set.
These disks cannot lie in the trap door since T
vanishes as . (Remember
is never in the trap door when n 2.)
230
Sketch of the proof
Suppose the Julia sets do not converge to the
unit disk D as
Then there exists and a sequence
such that, for each i, there is a point
such that lies in the
Fatou set.
The must accumulate on some nonzero
point, say , so we may assume that
lies in the Fatou set for all i.
231
Sketch of the proof
Suppose the Julia sets do not converge to the
unit disk D as
Then there exists and a sequence
such that, for each i, there is a point
such that lies in the
Fatou set.
The must accumulate on some nonzero
point, say , so we may assume that
lies in the Fatou set for all i.
232
Sketch of the proof
Suppose the Julia sets do not converge to the
unit disk D as
Then there exists and a sequence
such that, for each i, there is a point
such that lies in the
Fatou set.
The must accumulate on some nonzero
point, say , so we may assume that
lies in the Fatou set for all i.
But for large i, so stretches into an
annulus that surrounds the origin, so
this disconnects the Julia set.
233
So the Fatou components must become arbitrarily
small
234
For the family
the Julia sets again converge to the unit
disk, but only if ? ? 0 along n - 1 special
rays. (with M. Morabito)
n 6
n 4
235
Things are much different in the family
when 1/n 1/d lt 1, i.e. n, d ? 2 (but not
both 2)
236
n d 3 Note the round annuli in the Fatou
set
237
It can be shown that, when n, d 2 and not
both equal to 2 and ? lies in the McMullen
domain, the Fatou set always contains a round
annulus of some fixed width, so the Julia sets do
not converge to the unit disk in this case.
238

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

c 0
239

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

c -1
240

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

c -.12 .75i
241
When , the Julia set again expodes
and converges to the filled Julia set for z2
c. (with R. Kozma)
242
When , the Julia set again expodes
and converges to the filled Julia set for z2
c. (with R. Kozma)
243
When , the Julia set again expodes
and converges to the filled Julia set for z2
c. (with R. Kozma)
244
When , the Julia set again expodes
and converges to the filled Julia set for z2
c. (with R. Kozma)
245
When , the Julia set again expodes
and converges to the filled Julia set for z2
c. (with R. Kozma)
An inverted Douady rabbit
246
If you chop off the ears of each internal
rabbit in each component of the original Fatou
set, then whats left is another Sierpinski curve
(provided that both of the critical orbits
eventually escape).
247
The case n gt 2 is also very different
(E. Russell)
248
When ? is small, the Julia set contains a
Cantor set of circles surrounding the
origin.....
249
infinitely many of which are decorated and
there are also Cantor sets of buried points
250
Part 4 Cantor necklaces and webs

A Cantor necklace is the Cantor middle thirds
set with open disks replacing the removed
intervals.
251
Part 4 Cantor necklaces and webs

A Cantor necklace is the Cantor middle thirds
set with open disks replacing the removed
intervals.
Do you see a necklace in the carpet?
252
Part 4 Cantor necklaces and webs

A Cantor necklace is the Cantor middle thirds
set with open disks replacing the removed
intervals.
Do you see a necklace in the carpet?
253
Part 4 Cantor necklaces and webs

A Cantor necklace is the Cantor middle thirds
set with open disks replacing the removed
intervals.
There are lots of necklaces in the carpet
254
Part 4 Cantor necklaces and webs

A Cantor necklace is the Cantor middle thirds
set with open disks replacing the removed
intervals.
There are lots of necklaces in the carpet
255

There are lots of Cantor necklaces in these Julia
sets, just as in the Sierpinski carpet.
a Julia set with n 2
256

There are lots of Cantor necklaces in these Julia
sets, just as in the Sierpinski carpet.
another Julia set with n 2
257

There are lots of Cantor necklaces in these Julia
sets, just as in the Sierpinski carpet.
another Julia set with n 2
258

There are lots of Cantor necklaces in these Julia
sets, just as in the Sierpinski carpet.
another Julia set with n 2
259

There are lots of Cantor necklaces in these Julia
sets, just as in the Sierpinski carpet.
another Julia set with n 2
260
Even if we choose a parameter from the Mandelbrot
set, there are Cantor necklaces in the Julia
set
n 2
261
Even if we choose a parameter from the Mandelbrot
set, there are Cantor necklaces in the Julia
set
n 2
262
n 2
And there are Cantor necklaces in the parameter
planes.
263
The critical circle ray
264
The critical circle ray
Critical points
265
The critical circle ray
Critical points
0
n 2
266
The critical circle ray
Critical points
Critical values
0
n 2
267
The critical circle ray
Critical points
Critical values
0
n 2
268
The critical circle ray
Critical points
Critical values
Prepoles
0
n 2
269
The critical circle ray
Critical points
Critical values
Prepoles
0
n 2
270
The critical circle ray
Critical points
Critical values
Prepoles
0
Critical points prepoles lie on the critical
circle
n 2
271
The critical circle ray
Critical points
Critical values
Prepoles
0
Critical points prepoles lie on the critical
circle
n 2
272
The critical circle ray
Critical points
Critical values
Prepoles
0
Critical points prepoles lie on the critical
circle
Which is mapped 2n to 1 onto the critical line
n 2
273
The critical circle ray
Critical points
Critical values
Prepoles
0
Critical points prepoles lie on the critical
circle
Which is mapped 2n to 1 onto the critical line
n 2
274
Critical points
Critical values
Critical point rays
n 2
275
Critical points
Critical values
Critical point rays are mapped 2 to 1 to a ray
external to the critical line, a critical value
ray.
n 2
276
Suppose is not positive real.
n 2
277
Suppose is not positive real. Then the
critical values do not lie on the
critical point rays.
278
Suppose is not positive real.
There are 4 prepole sectors when n 2
and the critical values always lie in I1 and
I3.
I1
I2
I0
I3
279
Suppose is not positive real.
There are 4 prepole sectors when n 2
and the critical values always lie in I1 and
I3.
I1
I2
And the interior of each Ij is
mapped one-to-one over the entire plane minus the
critical value rays
I0
I3
280
Suppose is not positive real.
There are 4 prepole sectors when n 2
and the critical values always lie in I1 and
I3.
I1
I2
And the interior of each Ij is
mapped one-to-one over the entire plane minus the
critical value rays
I0
I3
In particular, maps I0 and I2 one-to-one
over I0 I2
281
Choose a circle in B that is mapped strictly
outside itself
I2
I0
282
Choose a circle in B that is mapped strictly
outside itself
I2
I0
283
Choose a circle in B that is mapped strictly
outside itself
Then there is another circle in the
trap door that is also mapped to
284
Consider the portions of I0 and I2 that lie
between and , say U0 and U2
U2
U0
285
Consider the portions of I0 and I2 that lie
between and , say U0 and U2
U2
U0 and U2 are mapped one-to-one over U0 U2

U0
286
Consider the portions of I0 and I2 that lie
between and , say U0 and U2
U2
U0 and U2 are mapped one-to-one over U0 U2

U0
So the set of points whose orbits lie for all
iterations in U0 U2 is an invariant Cantor set
287
Consider the portions of I0 and I2 that lie
between and , say U0 and U2
U0 and U2 are mapped one-to-one over U0 U2

So the set of points whose orbits lie for all
iterations in U0 U2 is an invariant Cantor set
288
The Cantor set in U0 ? U2 contains
2 points on ?B
289
The Cantor set in U0 ? U2 contains
?B
2 points on ?B
?B
290
The Cantor set in U0 ? U2 contains
2 points on ?B
2 points on ?T
?T
291
The Cantor set in U0 ? U2 contains
2 points on ?B
2 points on ?T
4 points on the 2 preimages of ?T
292
The Cantor set in U0 ? U2 contains
2 points on ?B
2 points on ?T
4 points on the 2 preimages of ?T
Add in the appropriate preimages of ?T.....
293
The Cantor set in U0 ? U2 contains
2 points on ?B
2 points on ?T
4 points on the 2 preimages of ?T
Add in the appropriate preimages of ?T..... to
get an invariant Cantor necklace in
the dynamical plane
294
Cantor necklaces in the dynamical plane when n
2
295
When n gt 2, get Cantor webs in the dynamical
plane
296
When n gt 2, get Cantor webs in the dynamical
plane
Start with an open disk....
n 3
297
When n gt 2, get Cantor webs in the dynamical
plane
Then surround it by 4 smaller disks
n 3
298
When n gt 2, get Cantor webs in the dynamical
plane
Then do it again....
n 3
299
When n gt 2, get Cantor webs in the dynamical
plane
and so forth, joining the open disks by a Cantor
set of points
n 3
300
When n gt 2, get Cantor webs in the dynamical
plane
n 4
n 3
301
When n gt 2, get Cantor webs in the dynamical
plane
n 3
n 4
302
When n gt 2, get Cantor webs in the dynamical
plane
n 3
n 3
303
Same argument we have 2n prepole sectors
I0,...,I2n-1
I0
I1
I2
I5
I3
I4
n 3
304
Same argument we have 2n prepole sectors
I0,...,I2n-1
If is not positive real, then the critical
value ray always lies in I0 and In as long
as lies in a symmetry region
I0
I1
I2
I5
I3
I4
n 3
305
Same argument we have 2n prepole sectors
I0,...,I2n-1
If is not positive real, then the critical
value ray always lies in I0 and In as long
as lies in a symmetry region
I0
I1
I2
I5
A symmetry region when n 3
I3
I4
n 3
306
Same argument we have 2n prepole sectors
I0,...,I2n-1
If is not positive real, then the critical
value ray always lies in I0 and In as long
as lies in a symmetry region
I0
I1
I2
I5
A symmetry region when n 4
I3
I4
n 3
307
Same argument we have 2n prepole sectors
I0,...,I2n-1
If is not positive real, then the critical
value ray always lies in I0 and In as long
as lies in a symmetry region
So consider the corresponding regions Uj not
including U0 and Un which contain
U1
U2
U5
U4
n 3
308
Each of these Uj are mapped univalently over
all the others, excluding U0 and Un, so we
get an invariant Cantor set in these regions.
U1
U2
U5
U4
n 3
309
Each of these Uj are mapped univalently over
all the others, excluding U0 and Un, so we
get an invariant Cantor set in these regions.
Then join in the preimages of T in these
regions....
n 3
310
Each of these Uj are mapped univalently over
all the others, excluding U0 and Un, so we
get an invariant Cantor set in these regions.
Then join in the preimages of T in these
regions....
and their preimages....
n 3
311
Each of these Uj are mapped univalently over
all the others, excluding U0 and Un, so we
get an invariant Cantor set in these regions.
Then join in the preimages of T in these
regions....
and their preimages....
etc., etc. to get the Cantor web
n 3
312
Other Cantor webs
n 4
n 5
313
Other Cantor webs
Next time well see how Cantor webs and
necklaces also appear in the parameter planes for
these maps.