Topological Structures in the Julia Sets
of Rational Maps
Dynamics of the family of complex maps

n
F (z)  z 
zn
with:
Paul Blanchard
Toni Garijo

Matt Holzer
U. Hoomiforgot
Dan Look
Sebastian Marotta
Mark Morabito
Monica Moreno Rocha
Kevin Pilgrim
Elizabeth Russell
Yakov Shapiro
David Uminsky
Three different types of topological objects:
1. Cantor Necklaces
A Cantor necklace is a planar set that is
homeomorphic to the Cantor middle thirds
set with open disks replacing removed intervals.
Three different types of topological objects:
1. Cantor Necklaces
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dynamical plane
n=2
Three different types of topological objects:
1. Cantor Necklaces
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dynamical plane
n=2
Three different types of topological objects:
1. Cantor Necklaces
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dynamical plane
n=2
Three different types of topological objects:
1. Cantor Necklaces
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parameter plane
dynamical plane
n=2
Three different types of topological objects:
2. Mandelpinski Necklaces
Infinitely many simple closed curves in the parameter plane
that pass alternately through centers of “Sierpinski holes”
and centers of baby Mandelbrot sets.
Three different types of topological objects:
2. Mandelpinski Necklaces
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parameter plane
zoom in
n=3
Three different types of topological objects:
3. CanManPinski Trees
A tree of Cantor necklaces with Mandelbrot
sets interspersed between each branch
Three different types of topological objects:
3. CanManPinski Trees
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parameter plane
parameter plane
n=2
Three different types of topological objects:
3. CanManPinski Trees
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parameter plane
parameter plane
n=2
Dynamics of
F (z)  z 
n

zn
 , z complex and n  2

The Julia set J(F ) 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 J(F ).
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

QuickTime™ and a
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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

QuickTime™ and a
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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle

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0
F (z)  z 
2

z2
When   0 , the Julia set
is the unit circle
But when   0 , the
Julia set explodes


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0
F (z)  z 
2

z2
  1/16
When   0 , the Julia set
is the unit circle
But when   0 , the
Julia set explodes


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0
F (z)  z 
2

z2
  1/16
A Sierpinski curve
When   0 , the Julia set
is the unit circle
But when   0 , the
Julia set explodes


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0
F (z)  z 
2

z2
  0.01
Another Sierpinski
curve
When   0 , the Julia set
is the unit circle
But when   0 , the
Julia set explodes


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0
F (z)  z 
2

z2
  0.2
Also a Sierpinski
curve
Sierpinski Curve
A Sierpinski curve is any planar
set that is homeomorphic to the
Sierpinski carpet fractal.
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The Sierpinski Carpet
F (z )  z 
3
Easy computations:
2n free critical points
c  1/2n

z3

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
  .036.026i
F (z )  z 
3
Easy computations:
2n free critical points
c  1/2n

z3

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
  .036.026i
F (z )  z 
3
Easy computations:
2n free critical points
c  1/2n
Only 2 critical values


z3

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v  2 

  .036.026i
F (z )  z 
3
Easy computations:
2n free critical points
c  1/2n
Only 2 critical values


z3

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v  2 

  .036.026i
F (z )  z 
3
Easy computations:
2n free critical points
c  1/2n
Only 2 critical values


z3

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v  2 

  .036.026i
F (z )  z 
3
Easy computations:
2n free critical points
c  1/2n
Only 2 critical values


z3

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v  2 
But really only 1 free
critical orbit since

the map has 2n-fold
symmetry
  .036.026i
F (z )  z 
3
Easy computations:
 is superattracting, so


z3
B
have immediate basin B
mapped n-to-1 to itself.
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  .036.026i
F (z )  z 
3
Easy computations:
 is superattracting, so


z3
B
have immediate basin B
mapped n-to-1 to itself.
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T
0 is a pole, so have
trap door T mapped
n-to-1 to B.
  .036.026i
F (z )  z 
3
Easy computations:
 is superattracting, so


z3
B
have immediate basin B
mapped n-to-1 to itself.
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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.
  .036.026i
The Escape Trichotomy
(with Dan Look & David Uminsky)
There are three distinct ways the critical orbit can enter B:
v  B

v  T



J(F ) is a Cantor set
J(F ) is a Cantor set of
simple closed curves
(this case does not occur if n = 2)


F (v ) T 

k


J(F ) is a Sierpinski curve
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parameter plane
when n = 3
Case 1:
v  B

J(F ) is a Cantor set
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
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
parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
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
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
parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
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
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
parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
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
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
parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
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
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
parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
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
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
parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
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
parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
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parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
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
parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
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
parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
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
parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
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
parameter plane
when n = 3
v  B

J is a Cantor set
J(F ) is a Cantor set
Case 2: the critical values lie in T, not B
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parameter plane
when n = 3

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
parameter plane
when n = 3
v  T

 lies in the McMullen domain

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
parameter plane
when n = 3
v  T

J is a Cantor set of
simple closed curves
 lies in the McMullen domain
Remark: There is no McMullen domain in the case n = 2.

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
parameter plane
when n = 3
v  T

J is a Cantor set of
simple closed curves
 lies in the McMullen domain

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
parameter plane
when n = 3
v  T

J is a Cantor set of
simple closed curves
 lies in the McMullen domain

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
parameter plane
when n = 3
v  T

J is a Cantor set of
simple closed curves
 lies in the McMullen domain

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
parameter plane
when n = 3
v  T

J is a Cantor set of
simple closed curves
 lies in the McMullen domain

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
parameter plane
when n = 3
v  T

J is a Cantor set of
simple closed curves
 lies in the McMullen domain

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
parameter plane
when n = 3
v  T

J is a Cantor set of
simple closed curves
 lies in the McMullen domain
Case 3: the critical orbit eventually lands in the trap door.
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parameter plane
when n = 3
F (v )  T
k

 lies in a Sierpinski hole

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole
1. Cantor necklaces in the
dynamical and parameter plane
The Cantor necklace is homeomorphic
to the Cantor middle thirds set with
open disks replacing removed intervals.
1. Cantor necklaces in the
dynamical and parameter plane
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The Cantor necklace is homeomorphic
to the Cantor middle thirds set with
open disks replacing removed intervals.
Julia set n = 2
 = -0.23
1. Cantor necklaces in the
dynamical and parameter plane
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The Cantor necklace is homeomorphic
to the Cantor middle thirds set with
open disks replacing removed intervals.
parameter plane
n=4
Dynamical plane: n = 2
B
T
Suppose B and T are disjoint.
Dynamical plane: n = 2
1/4
Four critical points
Dynamical plane: n = 2
21/2
And two critical values
that do not lie in T
Dynamical plane: n = 2
The critical lines...
Dynamical plane: n = 2
are mapped two-to-one to
one of two critical value rays
S1
S0
So the sectors S0 and S1 are mapped
one-to-one to C - {critical value rays)
Dynamical plane: n = 2
I1
I0
And the regions I0 - T and I1 - T are mapped
one-to-one to C - B - {critical value rays)
Dynamical plane: n = 2
And the regions I0 - T and I1 - T are mapped
one-to-one to C - B - {critical value rays)
Dynamical plane: n = 2
I1
T
I0
So consider the bow-tie I0  T  I1
Dynamical plane: n = 2
I1
T
I0
Both I0 and I1 are mapped one-to-one
over the entire bow-tie I0  T  I1
Dynamical plane: n = 2
T
So we have a preimage of the bow-tie
inside each of I0 and I1
Dynamical plane: n = 2
T
Then a second preimage, etc., etc.
Dynamical plane: n = 2
T
The points whose orbits stay in I0  I1 form a Cantor
set, and the preimages of T give the adjoined disks.
Dynamical plane: n = 2
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The points whose orbits stay in I0  I1 form a Cantor
set, and the preimages of T give the adjoined disks.
Cantor Necklaces in the Parameter Plane
F (z)  z 
2
c = 1/4

v = 2 1/2

z2
F(v) = 1/4 + 4
D = {| || < 1, Re() < 0}
D
For each   D, have a Cantor set of points inside I1
I1
... :. ... . . .
.
T
I0
For each   D, have a Cantor set of points inside I1
Let zs() be the point in the Cantor set with itinerary s
I1
... :. ... . . .
.
zs()
T
I0
For each   D, have a Cantor set of points inside I1
Let zs() be the point in the Cantor set with itinerary s
zs() depends analytically on 
and continuously on s
I1
... :. ... . . .
.
zs()
T
I0
For each   D, have a Cantor set of points inside I1
Let zs() be the point in the Cantor set with itinerary s
zs() depends analytically on 
and continuously on s
and zs() lies in the half-disk
H given by |z| < 2, Re(z) < 0
I1
... :. ... . . .
.
zs()
H
T
I0
So have an analytic map   zs()
that takes D into H
zs()
D
H
So have an analytic map   zs()
that takes D into H
Have another map   G() = F(v) = 1/4 + 4 which
maps D over a larger half disk containing H
zs()
G()
D
H
So have an analytic map   zs()
that takes D into H
Have another map   G() = F(v) = 1/4 + 4 which
maps D over a larger half disk containing H
But G is invertible.
zs()
G-1
D
H
So have an analytic map   zs()
that takes D into H
Have another map   G() = F(v) = 1/4 + 4 which
maps D over a larger half disk containing H
But G is invertible. So G-1(zs()) maps D strictly inside itself.
zs()
G-1
D
H
By the Schwarz Lemma, for each itinerary s there is a
unique fixed point s for the map G-1(zs()).
zs()
s
G-1
D
H
By the Schwarz Lemma, for each itinerary s there is a
unique fixed point s for the map G-1(zs()).
This is a parameter for which G(s) = zs(s),
i.e., the second iterate of the critical points lands
on a point in the Cantor set portion of the Cantor necklace.
zs()
s
zs(s)
G-1
D
H
So the points s for each s give a
Cantor set of points in the parameter plane.
zs()
s
zs(s)
G-1
D
H
So the points s for each s give a
Cantor set of points in the parameter plane.
Similar arguments involving Böttcher coordinates on
and itineraries of preimages of the trap door
then append the Sierpinski holes to the necklace.
zs()
s
zs(s)
G-1
D
H
This necklace lies along the negative real axis.
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parameter plane
n=2
There are lots of other Cantor
necklaces in the parameter planes.
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parameter plane
n=2
There are lots of other Cantor
necklaces in the parameter planes.
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parameter plane
n=2
There are lots of other Cantor
necklaces in the parameter planes.
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parameter plane
n=2
There are lots of other Cantor
necklaces in the parameter planes.
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parameter plane
n=2
When n > 2, get more complicated Cantor webs
case n = 3:
When n > 2, get more complicated Cantor webs
case n = 3:
When n > 2, get more complicated Cantor webs
case n = 3:
When n > 2, get more complicated Cantor webs
case n = 3:
When n > 2, get more complicated Cantor webs
case n = 3:
Continue in this way
and then adjoin
Cantor sets
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Dynamical plane
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Parameter plane n = 3
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Dynamical plane
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Parameter plane n = 3
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Dynamical plane
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Parameter plane n = 3
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Dynamical plane
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Parameter plane n = 3
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Dynamical plane
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Parameter plane n = 3
Cantor webs in the parameter plane
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Parameter plane n = 3
Cantor webs in the parameter plane
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Parameter plane n = 3
Cantor webs in the parameter plane
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Parameter plane
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Parameter plane n = 3
Cantor webs in the parameter plane
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Parameter plane
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Parameter plane n = 3
Cantor webs in the parameter plane
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Parameter plane
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Parameter plane n = 3
Different Cantor webs when n = 4
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Dynamical plane
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Parameter plane n = 4
Part 2: Mandelpinski Necklaces
A “Mandelpinski necklace” is a simple closed curve in the
parameter plane that passes alternately through k centers
of baby Mandelbrot sets and k centers of Sierpinski holes.
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Parameter plane for n = 3
A “Mandelpinski necklace” is a simple closed curve in the
parameter plane that passes alternately through k centers
of baby Mandelbrot sets and k centers of Sierpinski holes.
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C1 passes through the
centers of 2 M-sets
and 2 S-holes
Easy check: C1 is the
circle r = 2-2n/n-1
Parameter plane for n = 3
A “Mandelpinski necklace” is a simple closed curve in the
parameter plane that passes alternately through k centers
of baby Mandelbrot sets and k centers of Sierpinski holes.
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Parameter plane for n = 3
A “Mandelpinski necklace” is a simple closed curve in the
parameter plane that passes alternately through k centers
of baby Mandelbrot sets and k centers of Sierpinski holes.
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C2 passes through the
centers of 4 M-sets*
and 4 S-holes
* only exception:
2 centers of period
2 bulbs, not M-sets
Parameter plane for n = 3
A “Mandelpinski necklace” is a simple closed curve in the
parameter plane that passes alternately through k centers
of baby Mandelbrot sets and k centers of Sierpinski holes.
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Parameter plane for n = 3
C3 passes through the
centers of 10 M-sets
and 10 S-holes
A “Mandelpinski necklace” is a simple closed curve in the
parameter plane that passes alternately through k centers
of baby Mandelbrot sets and k centers of Sierpinski holes.
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Parameter plane for n = 3
C4 passes through the
centers of 28 M-sets
and 28 S-holes
A “Mandelpinski necklace” is a simple closed curve in the
parameter plane that passes alternately through k centers
of baby Mandelbrot sets and k centers of Sierpinski holes.
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Parameter plane for n = 3
C5 passes through the
centers of 82 M-sets
and 82 S-holes
Theorem: There exist closed curves Cj, j  1,...,
surrounding the McMullen domain. Each Cj passes
alternately through (n-2)nj-1 +1 centers of baby
Mandelbrot sets and centers of Sierpinski holes.

Theorem: There exist closed curves Cj, j  1,...,
surrounding the McMullen domain. Each Cj passes
alternately through (n-2)nj-1 +1 centers of baby
Mandelbrot sets and centers of Sierpinski holes.

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Parameter plane for n = 3
C14 passes through the
centers of 4,782,969
M-sets and S-holes
Theorem: There exist closed curves Cj, j  1,...,
surrounding the McMullen domain. Each Cj passes
alternately through (n-2)nj-1 +1 centers of baby
Mandelbrot sets and centers of Sierpinski holes.

C1: 3 holes and M-sets
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Parameter plane for n = 4
Theorem: There exist closed curves Cj, j  1,...,
surrounding the McMullen domain. Each Cj passes
alternately through (n-2)nj-1 +1 centers of baby
Mandelbrot sets and centers of Sierpinski holes.

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Parameter plane for n = 4
C2: 9 holes and M-sets*
C3: 33 holes and M-sets
F (z )  z 
3
Easy computations:
Critical points: 1/2n
Prepoles: (-)1/2n

z3

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  .08i
F (z )  z 
3
Easy computations:
All of the critical
points and prepoles
lie on the “critical
circle”  0 : |z| = | | 1/2n

z3
0



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
  .08i
F (z )  z 
3
Easy computations:
All of the critical
points and prepoles
lie on the “critical
circle”  0 : |z| = | | 1/2n


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v



z3
0
v
which is mapped 2n-to-1
 the “critical
onto
value line”

connecting v

  .08i
F (z )  z 
3
Easy computations:
Any other circle around 0
is mapped n-to-1 to an ellipse 
whose foci are 2 
z3
0
v



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v


  .08i
F (z )  z 
3
Easy computations:
Any other circle around 0
is mapped n-to-1 to an ellipse 
whose foci are 2 
z3
0
v



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v


  .08i
F (z )  z 
3
Easy computations:
Any other circle around 0
is mapped n-to-1 to an ellipse 
whose foci are 2 
So the exterior of  0 is mapped
as an n-to-1
 covering of the
exterior of the critical value line.


z3
0
v

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v


  .08i
F (z )  z 
3
Easy computations:
Any other circle around 0
is mapped n-to-1 to an ellipse 
whose foci are 2 
So the exterior of  0 is mapped
as an n-to-1
 covering of the
exterior of the critical value line.
Same with
 the interior of  0.
z3
0
v

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v




  .08i
Now assume that v lies inside the critical circle:

v
v

0

Warning: this is not a real proof....

Now assume that v lies inside the critical circle:
The exterior of  0 is mapped n-to-1
onto the exterior of the critical value
 is a preimage 
ray, so there
1
mappedn-to-1 to  0 ,

v

v

0


Now assume that v lies inside the critical circle:
The exterior of  0 is mapped n-to-1
onto the exterior of the critical value
 is a preimage 
ray, so there
1
mappedn-to-1 to  0 ,

v

v

0



1
Now assume that v lies inside the critical circle:
The exterior of  0 is mapped n-to-1
onto the exterior of the critical value
 is a preimage 
ray, so there
1
mappedn-to-1 to  0 ,
then  2 is mapped

n-to-1 to  1 ,
v



v

0


 
1 2
Now assume that v lies inside the critical circle:
The exterior of  0 is mapped n-to-1
onto the exterior of the critical value
 is a preimage 
ray, so there
1
mappedn-to-1 to  0 ,
then  2 is mapped

n-to-1 to  1 ,
v

and on and on
 out to B


v

0
1 2


 
B
 0 contains 2n critical points and 2n prepoles, so
 1 contains 2n2 pre-critical points and pre-prepoles
v
v

0



1
 0 contains 2n critical points and 2n prepoles, so
 1 contains 2n2 pre-critical points and pre-prepoles
 k contains 2nk+1 points that
map to the critical points
and pre-prepoles
under Fk
v
v


0
1 2


 
B
 0 contains 2n critical points and 2n prepoles, so
 1 contains 2n2 pre-critical points and pre-prepoles
 k contains 2nk+1 points that
map to the critical points
and pre-prepoles
under Fk
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
n=3
 0 contains 2n critical points and 2n prepoles, so
 1 contains 2n2 pre-critical points and pre-prepoles
 k contains 2nk+1 points that
map to the critical points
and pre-prepoles
under Fk
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
n=3
 0 contains 2n critical points and 2n prepoles, so
 1 contains 2n2 pre-critical points and pre-prepoles
 k contains 2nk+1 points that
map to the critical points
and pre-prepoles
under Fk
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
n=3
As  rotates by one turn, these 2nk+1 points on  k
each rotate by 1/2nk+1 of a turn.


v
v

0



k

As  rotates by one turn, these 2nk+1 points on  k
each rotate by 1/2nk+1 of a turn.

Since

1 1n/2
F (v )  2   n n/2  n 
2 
2
n
n/2
the second iterate of the critical
points rotate by 1 - n/2 of
a turn
v
v

0



k

As  rotates by one turn, these 2nk+1 points on  k
each rotate by 1/2nk+1 of a turn.

Since

1 1n/2
F (v )  2   n n/2  n 
2 
2
n
n/2
the second iterate of the critical
points rotate by 1 - n/2 of
a turn, so this point hits
exactly
(n / 2 1)(2n
k 1
) 1  (n  2)n
k 1
preimages of the critical points
and prepoles on  k
1
v
v

0



k
The real proof involves the Schwarz Lemma (as before):
There is a natural parametrization  k ( ) of each  k


 k ( )
v
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v


0



k
The real proof involves the Schwarz Lemma (as before):
There is a natural parametrization  k ( ) of each  k


 k ( )
v
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
Best to restrict to a “symmetry
region” inside the circle C1,


so that k ( ) is well-defined.
v

0




k
Then we have a second map from the parameter plane to the
dynamical plane, namely G( )  F (v ) which is invertible
on the symmetry sector

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 k ( )
G 1

Best to restrict to a “symmetry

region” inside the circle C1,


so that k ( ) is well-defined.
v
v

0




k
Then we have a second map from the parameter plane to the
dynamical plane, namely G( )  F (v ) which is invertible
on the symmetry sector

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G 1
So consider the composition

G 1 ( k ( ) )
a map from a “disk” to itself.

 k ( )
v
v

0



k
Schwarz implies that G 1 ( k ( ) ) has a unique fixed point,
i.e., a parameter for which the second iterate of the critical
point lands on the point  k ( ), so this proves the
existence of 
lots of parameters for which the critical
orbits are periodic and land on 0.

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G 1
So consider the composition

G 1 ( k ( ) )
a map from a “disk” to itself.

 k ( )
v
v

0



k
Remarks:
1. This proves the existence of centers of
Sierpinski holes and Mandelbrot sets. Producing
the entire M-set involves polynomial-like maps;
while the entire S-hole involves qc-surgery.
Remarks:
1. This proves the existence of centers of
Sierpinski holes and Mandelbrot sets. Producing
the entire M-set involves polynomial-like maps;
while the entire S-hole involves qc-surgery.
2. It is known that each S-hole in the Mandelpinski
necklace is also surrounded by infinitely many
sub-necklaces, which in turn are surrounded by
sub-sub-necklaces, etc.
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n=3
3. CanManPinski Trees
A tree of Cantor necklaces with Mandelbrot
sets interspersed between each branch
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parameter plane
parameter plane
n=2
Recall that we have a Cantor necklace in the
dynamical plane lying in I0  T  I1
I0
I1
Dynamical plane: n = 2
The regions I2 and I3 are mapped one-to-one over
I0  T  I1, so there are Cantor necklaces in I2 and I3
I2
I0
I1
I3
Dynamical plane: n = 2
The regions I2 and I3 are mapped one-to-one over
I0  T  I1, so there are Cantor necklaces in I2 and I3
I2
I0
I1
I3
Dynamical plane: n = 2
This necklace is mapped one-to-one onto
the original necklace.
This necklace is mapped one-to-one onto
the original necklace.
And so is the
bottom necklace
Now consider the regions Sj.
S1
S0
S2
S3
Now consider the regions Sj.
S0 is mapped two-to-one onto S0  S1
S1
S0
S2
S3
Now consider the regions Sj.
S0 is mapped two-to-one onto S0  S1
S1
Similarly, S1  S2  S3,
S2  S0  S1
and S3  S2  S3
S0
S2
S3
Assuming  lies in the upper half plane, the critical
values v lie in S0 and S2 (easy check)
S1
v
S0
S2
-v
S3
Assuming  lies in the upper half plane, the critical
values v lie in S0 and S2 (easy check)
So there is a region in S3 mapped one-to-one onto
S3.
S1
v
S0
S2
-v
S3
So there is a preimage of this Cantor necklace in S3
S1
v
S0
S2
-v
S3
So there is a preimage of this Cantor necklace in S3,
S1
v
S0
S2
-v
S3
So there is a preimage of this Cantor necklace in S3,
and then another preimage,
S1
v
S0
S2
-v
S3
So there is a preimage of this Cantor necklace in S3,
and then another preimage, and so on, yielding infinitely
many necklaces eventually mapping to the original
necklace. Looking like branches
of a tree....
S1
v
S0
S2
-v
S3
By symmetry, we have similar branches in S1
S1
v
S0
S2
-v
S3
By symmetry, we have similar branches in S1, S0,
S1
S0
S2
S3
By symmetry, we have similar branches in S1, S0, and S2
S1
S0
S2
S3
This produces trees of Cantor necklaces in the dynamical plane
This produces trees of Cantor necklaces in the dynamical plane
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Assuming  is in the upper half plane, we can again use
G() = 1/4 + 4  and an appropriate coding of points in the
necklace, and then the Schwarz Lemma produces a similar
tree in the upper half of the parameter plane.
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Symmetry under complex conjugation yields a
similar tree in the lower half-plane.
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Then polynomial-like map theory produces a
Mandelbrot set in each region in between the branches.
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Open problems:
Can we use these trees to give a complete map of the
Sierpinski-hole regions and buried parameters
in the parameter planes?
Same question for the baby Mandelbrot sets.
Using “Yoccoz puzzles,” do these trees allow us to
see that the boundary of the parameter plane locus
is a simple closed curve?
Open problems:
Can we use these trees to give a complete map of the
Sierpinski-hole regions and buried parameters
in the parameter planes?
Same question for the baby Mandelbrot sets.
Using “Yoccoz puzzles,” do these trees allow us to
see that the boundary of the parameter plane locus
is a simple closed curve?
Would it be better to call these things
Cantormandelbrotsierpinski trees?
Open problems:
Can we use these trees to give a complete map of the
Sierpinski-hole regions and buried parameters
in the parameter planes?
Same question for the baby Mandelbrot sets.
Using “Yoccoz puzzles,” do these trees allow us to
see that the boundary of the parameter plane locus
is a simple closed curve?
Would it be better to call these things
Cantormandelbrotsierpinski trees?
Who the hell is this?
Parameter plane (rotated)
when n = 2
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Other topics:
Main cardioid of a buried baby M-set
Perturbed rabbit
Convergence to the unit disk
Major application
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
If  lies in the main cardioid of a buried Mandelbrot
set, then again the Julia set is a Sierpinski curve.
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A Sierpinski curve, but very different
dynamically from the earlier ones.
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n=4
Consider the family of maps
F (z)  z  c 
n

zn
where c is the center of a hyperbolic component
of the Mandelbrot set.

Consider the family of maps
F (z)  z  c 
n

zn
where c is the center of a hyperbolic component
of the Mandelbrot set.

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c 1
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0
Consider the family of maps
F (z)  z  c 
n

zn
where c is the center of a hyperbolic component
of the Mandelbrot set.

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c .12.75i
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0
When
  0, the Julia set again expodes.

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When
  0, the Julia set again expodes.

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When
  0, the Julia set again expodes.

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When
  0, the Julia set again expodes.

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When
  0, the Julia set again expodes.

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A doubly-inverted
Douady rabbit.
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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).
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The case n = 2 is very different from (and
much more difficult than) the case n > 2.
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n=3
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n=2
One difference: there is a McMullen domain when
n > 2, but no McMullen domain when n = 2
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n=3
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n=2
One difference: there is a McMullen domain when
n > 2, but no McMullen domain when n = 2
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n=3
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n=2
There is lots of structure when n > 2,
but what is going on when n = 2?
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n=3
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n=2
There is lots of structure when n > 2,
but what is going on when n = 2?
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TIFF (LZW) decompressor
are needed to see this picture.
n=3
QuickTime™ and a
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n=2
There is lots of structure when n > 2,
but what is going on when n = 2?
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TIFF (LZW) decompressor
are needed to see this picture.
n=3
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n=2
Also, not much is happening for the
Julia sets near 0 when n > 2
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n=3
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 .01
The Julia set is always a
Cantor set of circles.
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n=3
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 .0001
The Julia set is always a
Cantor set of circles.
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n=3
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 .000001
The Julia set is always a
Cantor set of circles.
There is always a round annulus
of some fixed width in the Fatou set,
so the Julia set does not converge
to the unit disk.
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 .000001
But when n = 2, lots of things happen near the origin;
in fact, the Julia sets converge to the unit disk as  0

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disk-converge
n=2
Here’s the parameter plane when n = 2:
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Rotate it by 90 degrees:
and this object appears everywhere.....
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