Structures in the Parameter Planes
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
Monica Moreno Rocha
Elizabeth Russell
Yakov Shapiro
David Uminsky
First a brief advertisement:
AIMS Conference on Dynamical Systems,
Differential Equations and Applications
Dresden University of Technology
Dresden, Germany
May 25-28 2010
Organizers: Janina Kotus, Xavier Jarque, me
One half hour slots for speakers.
Interested in attending/speaking?
Contact me at bob@bu.edu
Structures in the Parameter Planes
Dynamics of the family of complex maps

n
F (z)  z 
zn
What you see in the dynamical plane often reappears
(enchantingly so) in the parameter plane....

Cantor Necklaces:
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A Cantor necklace in a
Julia set when n = 2
Cantor Necklaces:
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A Cantor necklace in a
Julia set when n = 2
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and in the
parameter plane
Cantor Necklaces:
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A Cantor necklace in a
Julia set when n = 2
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and in the
parameter plane
Cantor Necklaces:
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A Cantor necklace in a
Julia set when n = 2
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and in the
parameter plane
Mandelpinski Necklaces:
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Circles of preimages of
the trap door and critical
points around 0
Mandelpinski Necklaces:
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Circles of preimages of
the trap door and critical
points around 0
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Circles through centers of
Sierpinski holes and baby
M -sets in the param-plane
Mandelpinski Necklaces:
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Circles of pre-preimages of
the trap door and pre-critical
points around 0
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Circles through centers of
Sierpinski holes and baby
M*-sets in the param-plane
* the only exception
Mandelpinski Necklaces:
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Circles of pre-preimages of
the trap door and pre-critical
points around 0
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Circles through centers of
Sierpinski holes and baby
M*-sets in the param-plane
* the only exception
Mandelpinski Necklaces:
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Circles of pre-preimages of
the trap door and pre-critical
points around 0
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Circles through centers of
Sierpinski holes and baby
M -sets in the param-plane
Mandelpinski Necklaces:
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Circles of pre-preimages of
the trap door and pre-critical
points around 0
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Circles through centers of
Sierpinski holes and baby
M -sets in the param-plane
Mandelpinski Necklaces:
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Circles of pre-preimages of
the trap door and pre-critical
points around 0
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Circles through centers of
Sierpinski holes and baby
M -sets in the param-plane
Mandelpinski Necklaces:
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Circles of pre-preimages of
the trap door and pre-critical
points around 0
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Circles through Sierpinski
holes and baby Mandelbrot
sets in the parameter plane
As Douady often said “You sow the seeds in the
dynamical plane and reap the harvest in the
parameter plane.”
It is often easy to prove something in the dynamical
plane, but harder to reproduce it in the parameter plane.
Here is how we will do this:
Suppose you have some object in the dynamical
plane that varies analytically with the parameter 
maybe a closed curve, maybe a Cantor necklace, or:

dynamical plane
Suppose you have some object in the dynamical
plane that varies analytically with the parameter 
maybe a closed curve, maybe a Cantor necklace, or:
Maybe it’s your face,
so call it Face(  )


dynamical plane
Suppose you have some object in the dynamical
plane that varies analytically with the parameter 
maybe a closed curve, maybe a Cantor necklace, or:
Maybe it’s your face,
so call it Face(  )

Change  , and Face(  )
moves analytically:



dynamical plane
Suppose you have some object in the dynamical
plane that varies analytically with the parameter 
maybe a closed curve, maybe a Cantor necklace, or:
Maybe it’s your face,
so call it Face(  )

Change  , and Face(  )
moves analytically:
maybe like this



dynamical plane
Suppose you have some object in the dynamical
plane that varies analytically with the parameter 
maybe a closed curve, maybe a Cantor necklace, or:
Maybe it’s your face,
so call it Face(  )

Change  , and Face(  )
moves analytically:
or like this (you’re so

cute!)


dynamical plane
So any particular point in Face(  ), say the tip of
your nose, nose(  ), varies analytically with 
nose( )



dynamical plane
So any particular point in Face(  ), say the tip of
your nose, nose(  ), varies analytically with 
nose( )



dynamical plane
So any particular point in Face(  ), say the tip of
your nose, nose(  ), varies analytically with 
nose( )



dynamical plane
So we have an analytic function nose(  ) from
parameter plane to the dynamical plane
nose(
 )

parameter plane
dynamical plane
So we have an analytic function nose(  ) from
parameter plane to the dynamical plane
nose(
 )



parameter plane
dynamical plane
Now suppose we have another analytic function G(  )
taking parameter plane to dynamical plane one-to-one
nose( )




G
parameter plane
dynamical plane
So we have an inverse map G-1 taking the dynamical
plane back to the parameter plane
nose( )



G-1
parameter plane
dynamical plane
Now suppose G takes a compact disk D in the parameter
plane to a disk in dynamical plane, and nose( ) is always
contained strictly inside G(D) when  D .
nose( )





D
parameter plane
G-1
G(D)
dynamical plane
So G-1(nose( )) maps D strictly inside itself.

nose( )



D
parameter plane
G-1
G(D)
dynamical plane
So G-1(nose( )) maps D strictly inside itself. So by
the Schwarz Lemma, there is a unique fixed point *
for the map G-1(nose(  )).

nose( * )


*


D
parameter plane
G-1
G(D)
dynamical plane
* is the unique parameter for which G( * ) = nose( * ).

*
nose(  )

*


D
parameter plane
G-1
G(D)
dynamical plane
* is the unique parameter for which G( * ) = nose( * ).
If we do this for each point in Face( ), we then
get the same “object” in the parameter plane.




G(D)
D
parameter plane
dynamical plane
* is the unique parameter for which G( * ) = nose( * ).
If we do this for each point in Face( ), we then
get the same “object” in the parameter plane.




G(D)
D
Why are you so unhappy
living in the parameter plane?
parameter plane
dynamical plane
The goal today is to show the existence in the
parameter plane of:
1.
2.
3.
4.
Cantor necklaces
Cantor webs
Mandelpinski necklaces
Cantor sets of circles of Sierpinski
curve Julia sets
1. Cantor Necklaces
A Cantor necklace is the Cantor
middle thirds set with open disks
replacing the removed intervals.
1. Cantor Necklaces
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A Cantor necklace is the Cantor
middle thirds set with open disks
replacing the removed intervals.
a Julia set with n = 2 and
a Cantor necklace
1. Cantor Necklaces
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A Cantor necklace is the Cantor
middle thirds set with open disks
replacing the removed intervals.
a Julia set with n = 2 and
another Cantor necklace
1. Cantor Necklaces
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TIFF (LZW) decompressor
are needed to see this picture.
A Cantor necklace is the Cantor
middle thirds set with open disks
replacing the removed intervals.
a Julia set with n = 2 and
lots of Cantor necklaces
1. Cantor Necklaces
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n=2
And there are Cantor necklaces in the parameter planes.
1. Cantor Necklaces
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n=2
1. Cantor Necklaces
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n=2
We’ll just show the existence of this Cantor necklace
along the negative real axis when n = 2.
Recall:
B = immediate basin of
T = trap door
F (z)  z 
3


z3
B


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T
Recall:
B = immediate basin of
T = trap door
2n free critical points
c  1/2n
F (z)  z 
3

z3
B

c



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T
Recall:
B = immediate basin of
T = trap door
2n free critical points
c  1/2n
F (z)  z 
3



v  2 
z3
B

c

Only 2 critical values

v
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T
Recall:
B = immediate basin of
T = trap door
2n free critical points
c  1/2n

Only 2 critical values


v  2 
2n prepoles
p  ( )1/2n
F (z)  z 
3



z3
B
p
c
v
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0
p
Consider

F (z)  z 
2

z
2
, 0
Consider
F (z)  z 
2

z
2
, 0
Since   0 , F preserves the real line.



Consider
F (z)  z 
2

z
2
, 0

need a glass
of wine???
graph of F (x)
Consider
F (z)  z 
2

z
2
, 0
B = basin of infinity

graph of F (x)
Consider
F (z)  z 
2

z
2
, 0
B = basin of infinity
T = trap door 
graph of F (x)
Consider
F (z)  z 
2

z
2
, 0
B = basin of infinity
T = trap door 
The two intervals I0 and
I1 are expanded over the
union of these intervals
and the trap door.
I1
I0
graph of F (x)
Consider
F (z)  z 
2

z
2
, 0
B = basin of infinity
T = trap door 
The two intervals I0 and
I1 are expanded over the
union of these intervals
and the trap door.
So there is an invariant Cantor
set on the negative real axis.
I1
I0
graph of F (x)
Consider
F (z)  z 
2

z
2
, 0
B = basin of infinity
T = trap door 
The two intervals I0 and
I1 are expanded over the
union of these intervals
and the trap door.
So there is an invariant Cantor
set on the negative real axis.
Add in the preimages of T
to get the Cantor necklace in
the dynamical plane for   0 .
I1
I0
graph of F (x)
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The Cantor necklace for negative 

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This portion is also a Cantor necklace
lying on the negative real axis for   0
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And we have a similar Cantor necklace
lying on the negative real axis in the
parameter plane for n = 2.
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
D
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
Let G( ) be the second iterate of the critical point
G()  F (v )


D
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
Let G( ) be the second iterate of the critical point

G( )  F (v )  (2  ) 
(2  )2
2


D
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
Let G( ) be the second iterate of the critical point

1
G( )  F (v )  (2  ) 
 4 
2
4
(2  )
2


D
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
Let G( ) be the second iterate of the critical point

1
G( )  F (v )  (2  ) 
 4 
2
4
(2  )
2


So G is 1-to-1 on D,
and maps D over itself;
G
D
-3.75
.25
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
Let G( ) be the second iterate of the critical point

1
G( )  F (v )  (2  ) 
 4 
2
4
(2  )
2


So G is 1-to-1 on D,
and maps D over itself;
equivalently, G-1 contracts
G(D) inside itself.
G-1
D
-3.75
.25
For any  in D (not just   0),
we also have an invariant Cantor
set as we showed earlier:


For any  in D (not just   0),
we also have an invariant Cantor
set as we showed earlier:
U and U are
portions
0
2
of a prepole sector
U2
c
U0

For any  in D (not just   0),
we also have an invariant Cantor
set as we showed earlier:
U and U are
portions
0
2
of a prepole sector
that are each mapped
univalently over both
U0 and U2.
U2
c
U0

For any  in D (not just   0),
we also have an invariant Cantor
set as we showed earlier:
U and U are
portions
0
2
of a prepole sector
that are each mapped
univalently over both
U0 and U2.
So there is a portion of
a Cantor set lying in U2.
c
U0

For any  in D (not just   0),
we also have an invariant Cantor
set as we showed earlier:
U and U are
portions
0
2
of a prepole sector
that are each mapped
univalently over both
U0 and U2.
So there is a portion of
a Cantor set lying in U2.
And we can add in the
appropriate preimages of
the trap door to get a
Cantor necklace.
c
U0

For any  in D (not just   0),
we also have an invariant Cantor
set as we showed earlier:
U and U are
portions
0
2
of a prepole sector
that are each mapped
univalently over both
U0 and U2.
So there is a portion of
a Cantor set lying in U2.
And we can add in the
appropriate preimages of
the trap door to get a
Cantor necklace.
G(D)
And, since  lies in D, the
Cantor set lies inside G(D).
We can identify each point in the Cantor set in U2 by
a unique sequence of 0’s and 2’s: s = (2 s1 s2 s3 ....)
given by the itinerary of the point.
U2
U0
We can identify each point in the Cantor set in U2 by
a unique sequence of 0’s and 2’s: s = (2 s1 s2 s3 ....)
given by the itinerary of the point.
So, for each such sequence s,
we have a map   zs (),
which is defined on D and
depends analytically on 
zs ( )



U2
U0
We therefore have two maps defined on D:
G(D)
D
We therefore have two maps defined on D:
1
1. The univalent map G( )  4  
4

G
D
G(D)
We therefore have two maps defined on D:
1
1. The univalent map G( )  4  
4
2. The point in the Cantor set zs ( )


G
D
zs
G(D)
1
So G ozs maps D strictly inside itself;

G-1
D
zs
G(D)
1
So G ozs maps D strictly inside itself; by
the Schwarz Lemma, there is a unique fixed point
*s in D for this map.


G-1
*s
D

zs
G(D)


1
So G ozs maps D strictly inside itself; by
the Schwarz Lemma, there is a unique fixed point
*s in D for this map.
*
*
G(

)

z
(

For this parameter, we have
s
s
s ), so
this is the unique parameter for which the critical
orbit lands on the point zs ( ) .


G-1
*s
D

zs
G(D)
*
This produces a Cantor set of parameters s ,
one for each sequence s.
Claim: this Cantor set lies on the negative real axis.

*
This produces a Cantor set of parameters s ,
one for each sequence s.
Claim: this Cantor set lies on the negative real axis.

Recall:
G()  4 1/ 4 , so G decreases from
.25 to -3.75 as  goes from 0 to -1 in D.


*
This produces a Cantor set of parameters s ,
one for each sequence s.
Claim: this Cantor set lies on the negative real axis.

Recall:
G()  4 1/ 4 , so G decreases from
.25 to -3.75 as  goes from 0 to -1 in D.

the Cantor set in the dynamical plane
lies on the negative real axis when   0.


*
This produces a Cantor set of parameters s ,
one for each sequence s.
Claim: this Cantor set lies on the negative real axis.

Recall:
G()  4 1/ 4 , so G decreases from
.25 to -3.75 as  goes from 0 to -1 in D.

the Cantor set in the dynamical plane
lies on the negative real axis when   0.

So G( ) must hit each point in the Cantor set along
the negative axis at least once.


*
This produces a Cantor set of parameters s ,
one for each sequence s.
Claim: this Cantor set lies on the negative real axis.

Recall:
G()  4 1/ 4 , so G decreases from
.25 to -3.75 as  goes from 0 to -1 in D.

the Cantor set in the dynamical plane
lies on the negative real axis when   0.

So G( ) must hit each point in the Cantor set along
the negative axis at least once.


So each parameter  in the parameter plane necklace
must also lie in [-1, 0]. This produces the Cantor set
portion of the necklace on the negative real axis.
*
s
Similar arguments produce parameters on
the negative axis that land after a specified
itinerary on a particular point in B (that
is determined by the Böttcher coordinate).
Similar arguments produce parameters on
the negative axis that land after a specified
itinerary on a particular point in B (that
is determined by the Böttcher coordinate).
And then these intervals can be expanded
to get the Sierpinski holes in the necklace.
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2. Cantor webs
Recall that, when n > 2, we have Cantor
“webs” in the dynamical plane:
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n=3
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n=4
2. Cantor webs
Recall that, when n > 2, we have Cantor
“webs” in the dynamical plane:
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n=3
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n=4
2. Cantor webs
Recall that, when n > 2, we have Cantor
“webs” in the dynamical plane:
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n=3
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n=3
2. Cantor webs
When n > 2, we also have Cantor
“webs” in the parameter plane:
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n=3
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n=4
2. Cantor webs
When n > 2, we also have Cantor
“webs” in the parameter plane:
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n=3
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n=4
2. Cantor webs
When n > 2, we also have Cantor
“webs” in the parameter plane:
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n=3
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n=4
A slightly different argument as in the case of the
Cantor necklaces works here. Say n = 3.
In the dynamical plane, we
had the disks Uj.
U1
v
U2
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
 v
U4

n=3
U5
A slightly different argument as in the case of the
Cantor necklaces works here. Say n = 3.
In the dynamical plane, we
had the disks Uj.
U1
U0
v
U2
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
 v
U4
U3

n=3
U5
Each of these Uj were mapped
univalently over all the others,
excluding U0 and Un, so
we found an invariant Cantor
set in these regions.
A slightly different argument as in the case of the
Cantor necklaces works here. Say n = 3.
In the dynamical plane, we
had the disks Uj.
U1
U0
v
U2
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
 v
U4
U3

n=3
U5
Each of these Uj were mapped
univalently over all the others,
excluding U0 and U3, so
we found an invariant Cantor
set in these regions.
U0 and U3 are mapped
univalently over these Uj,
so there is a preimage of
this Cantor set in both U0
and U3
Now let G( ) be one of the two critical values, so
G( )  2 
And
 choose a disk D in one
of the “symmetry sectors”

in the parameter
plane:
D
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U2
 v
U3
U4
Now let G( ) be one of the two critical values, so
G( )  2 
And
 choose a disk D in one
of the “symmetry sectors”

in the parameter
plane:
Then G maps D univalently
over all of U0, so we again get
a copy of the Cantor set in D
G
U1
U0
v
D
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U2
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
 v
U3
U4
U5
Now let G( ) be one of the two critical values, so
G( )  2 
And
 choose a disk D in one
of the “symmetry sectors”

in the parameter
plane:
Then G maps D univalently
over all of U0, so we again get
a copy of the Cantor set in D
G
U1
U0
D
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v
U2
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
Then adjoining the appropriateSierpinski
holes
v
gives a Cantor web in the parameter plane.
U4
U
3
U5
3. “Mandelpinski” necklaces
A Mandlepinski necklace is a simple closed curve in
the parameter plane that passes alternately through a
certain number of centers of baby M-sets and the
same number of centers of S-holes.
3. “Mandelpinski” necklaces
A Mandlepinski necklace is a simple closed curve in
the parameter plane that passes alternately through a
certain number of centers of baby M-sets and the
same number of centers of S-holes.
oops, sorry....
3. “Mandelpinski” necklaces
A Mandlepinski necklace is a simple closed curve in
the parameter plane that passes alternately through a
certain number of centers of baby M-sets and the
same number of centers of Sierpinski-holes.
3. “Mandelpinski” necklaces
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parameter plane n = 4
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A Julia set
3. “Mandelpinski” necklaces
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parameter plane n = 4
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There is a “ring” around T
passing through 8 = 2*4
preimages of T
3. “Mandelpinski” necklaces
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parameter plane n = 4
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There is a “ring” around T
passing through 8 = 2*4
preimages of T
3. “Mandelpinski” necklaces
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parameter plane n = 4
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Another “ring” around T
passing through 32 = 2*42
preimages of T
3. “Mandelpinski” necklaces
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parameter plane n = 4
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Another “ring” around T
passing through 32 = 2*42
preimages of T
3. “Mandelpinski” necklaces
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parameter plane n = 4
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Another “ring” around T
passing through 128 = 2*43
preimages of T
Now look around the McMullen domain in the parameter plane:
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parameter plane for n = 4
Now look around the McMullen domain in the parameter plane:
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There is a ring around M that passes alternately
through the centers of
3 = 2*40 + 1 Sierpinski holes and 3 Mandelbrot sets
Now look around the McMullen domain in the parameter plane:
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There is a ring around M that passes alternately
through the centers of
3 = 2*40 + 1 Sierpinski holes and 3 Mandelbrot sets
Now look around the McMullen domain in the parameter plane:
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Another ring around M that passes alternately
through the centers of
9 = 2*41 + 1 Sierpinski holes and 9 “Mandelbrot sets”*
*well, 3 period 2 bulbs
Now look around the McMullen domain in the parameter plane:
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Another ring around M that passes alternately
through the centers of
9 = 2*41 + 1 Sierpinski holes and 9 “Mandelbrot sets”*
*well, 3 period 2 bulbs
Now look around the McMullen domain in the parameter plane:
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Then 33 = 2*42 + 1 Sierpinski holes and 33 Mandelbrot sets
Now look around the McMullen domain in the parameter plane:
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Then 129 = 2*43 + 1 Sierpinski holes and 129 Mandelbrot sets
Similar kinds of rings occur in the other parameter planes:
n=3
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parameter plane for n = 3
Similar kinds of rings occur in the other parameter planes:
n=3
S0
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S0: 2 = 1*30 + 1 Sierpinski holes & M-sets
n=3
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S1: 4 = 1*31 + 1 Sierpinski holes & M-sets*
*well, two period 2 bulbs
n=3
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S2: 10 = 1*32 + 1 Sierpinski holes & “M-sets”
n=3
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S3: 28 = 1*33 + 1 Sierpinski holes & M-sets
n=3
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82, 244, then 730 Sierpinski holes...
n=3
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sorry, I forgot.....
nevermind
the 13th ring passes through
1,594,324 Sierpinski holes...
Theorem: For each n > 2, the McMullen domain is
surrounded by infinitely many simple closed curves Sk
(“Mandelpinski” necklaces) having the property that:
1. each Sk surrounds the McMullen domain and Sk+1,
and the Sk accumulate on the boundary of M;
2. Sk meets the center of exactly (n-2)nk-1 + 1 Sierpinski
holes, each with escape time k + 2;
3. Sk also passes through the centers of the same number
of baby Mandelbrot sets*
* with one exception
The critical points and
prepoles all lie on the
1/2n
“critical circle” r  |  |
c
p
p
c

The critical points and
prepoles all lie on the
1/2n
“critical circle” r  |  |
The critical circle is
mapped
 2n-to-1 onto
the “critical value ray”
c
p
p
v
0
c
The critical points and
prepoles all lie on the
1/2n
“critical circle” r  |  |
The critical circle is
mapped
 2n-to-1 onto
the “critical value ray”
And every other circle centered
at the origin and outside the
critical circle is mapped n-to-1
to an ellipse with foci at the critical
values
v
0
The critical points and
prepoles all lie on the
1/2n
“critical circle” r  |  |
The critical circle is
mapped
 2n-to-1 onto
the “critical value ray”
And every other circle centered
at the origin and outside the
critical circle is mapped n-to-1
to an ellipse with foci at the critical
values
v
0
The critical points and
prepoles all lie on the
1/2n
“critical circle” r  |  |
The critical circle is
mapped
 2n-to-1 onto
the “critical value ray”
And every other circle centered
at the origin and outside the
critical circle is mapped n-to-1
to an ellipse with foci at the critical
values, and same inside
v
0
There are no critical points
outside the critical circle,
so this region is mapped as
n-to-1 covering onto the
complement of the
critical value ray.
v
0
There are no critical points
outside the critical circle,
so this region is mapped as
n-to-1 covering onto the
complement of the
critical value ray.
The interior of the
critical circle is
also mapped
n-to-1 onto the
complement of the
critical value ray
v
0
The dividing circle contains all parameters for which
the critical values lie on the critical circle, i.e.,
|  |1/2n  2 |  |1/2  |  |  2 2n /(n1)

The dividing circle contains all parameters for which
the critical values lie on the critical circle, i.e.,
|  |1/2n  2 |  |1/2  |  |  2 2n /(n1)

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n  4 : |  |  2 8 / 3
When n = 4, the dividing
circle passes through 3
centers of Sierpinski holes
and 3 baby Mandelbrot sets
The dividing circle passes through n-1 centers of Sierpinski
holes and n-1 centers of baby Mandelbrot sets.
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n  4 : |  |  2 8 / 3
When n = 4, the dividing
circle passes through 3
centers of Sierpinski holes
and 3 baby Mandelbrot sets
Reason:

v
8 / 3
|

|

2

parameter plane n = 4


r  |  |8
dynamical plane
Reason: as  runs once around the dividing circle,


v
8 / 3
|

|

2

parameter plane n = 4


r  |  |8
dynamical plane
Reason: as  runs once around the dividing circle, v
rotates 1/2 of a turn,



v
8 / 3
|

|

2

parameter plane n = 4


r  |  |8
dynamical plane
Reason: as  runs once around the dividing circle, v
rotates 1/2 of a turn, while the critical points and prepoles
each rotate on 1/8 of a turn.



v
8 / 3
|

|

2

parameter plane n = 4


r  |  |8
dynamical plane
Reason: as  runs once around the dividing circle, v
rotates 1/2 of a turn, while the critical points and prepoles
each rotate on 1/8 of a turn.
So
 v meets 3 prepoles and critical points enroute.



v
8 / 3
|

|

2

parameter plane n = 4


r  |  |8
dynamical plane
So the ring S0 is just the dividing circle in parameter plane.
S0
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n=4
So the ring S0 is just the dividing circle in parameter plane.
For the other rings, let’s consider for simplicity
only the case where n = 4
S0
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n=4
When  lies inside the dividing circle, we have
| v |  2 | |  |  |  |1/8  | c |


When  lies inside the dividing circle, we have
| v |  2 | |  |  |  |1/8  | c |

so F maps the critical circle C0 strictly inside itself


v

v

C0
Now there is a preimage C1 of the critical circle that
is mapped 4-to-1 onto the critical circle, and this curve
contains 32 pre-critical points and 32 pre-pre-poles.
v

v

C0
C1
And then a preimage C2 of the C1 that is mapped 4-to-1
onto the C1, and so 16-to-1 onto C0, and this curve contains
128 pre-pre-critical points and 128 pre-pre-pre-poles, etc.
v

v
C0
C1
C2

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The rings C0 and C1
Let G( ) be the second iterate of the critical point
G( )  F (c )  2 
2


n
n/2

1
2n (n/2)1
Let G( ) be the second iterate of the critical point
G( )  F (c )  2 
2
n
n/2

1
So G( )  16  
16 
2



1
2n (n/2)1
when n = 4.
Let G( ) be the second iterate of the critical point
G( )  F (c )  2 
2
n
n/2

1
So G( )  16  
16 
2


1
2n (n/2)1
when n = 4.
Note that G()  as   0 provided n > 2.

1
When n = 2, G( )  4   , a very different situation.
4



G maps points in the parameter plane to
points in the dynamical plane
G
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C0
the critical circle
Let D be the open disk of radius 1/8 in the parameter plane.
G maps D univalently onto a region in the exterior of C0
G(D)
G
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D
C0
and G(D) covers C1, C2,...
G
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D
C0
Choose a small disk D0 inside M. Then G maps the
annulus A = D - D0 univalently over all of the Cj, j > 0.
G
A
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D
C0
k

Choose a parametrization of Ck, say  ( ). So we
k
have a second map from A into G(A),     ( )


G
A
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D
C0
 k ( )

Since G is 1-to-1, we thus have a map
H ( )  G 1 ( k ( )) which takes A into A.

A
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H
D
C0
 k ( )

Let S be the covering strip of A and let H*: S
be the covering map of H: A A
A
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S
H
D
C0
 k ( )

Let S be the covering strip of A and let H*: S
be the covering map of H: A A
S
By the Schwarz Lemma, for each given k,  , and  ,
k
there is a unique fixed point  ( ) for H* in A,
which depends analytically on  .




Let S be the covering strip of A and let H*: S
be the covering map of H: A A
S
By the Schwarz Lemma, for each given k,  , and  ,
k
there is a unique fixed point  ( ) for H* in A,
which depends analytically on  .


So the map    () gives a parametrization of the
ring Sk in the parameter plane,
and -values that

correspond to pre-poles or pre-critical points are then
parameters at the centers of Sierpinski holes or baby

Mandelbrot sets.

k
There are (n - 2)nk-1 pre-poles in the kth dynamical plane
ring, but (n - 2)nk-1 + 1 centers of Sierpinski holes in
the parameter plane rings. Here’s the reason:
There are (n - 2)nk-1 pre-poles in the kth dynamical plane
ring, but (n - 2)nk-1 + 1 centers of Sierpinski holes in
the parameter plane rings. Here’s the reason:
1
1

On the annulus A, G( )  16  
16  16 
2

There are (n - 2)nk-1 pre-poles in the kth dynamical plane
ring, but (n - 2)nk-1 + 1 centers of Sierpinski holes in
the parameter plane rings. Here’s the reason:
1
1

On the annulus A, G( )  16  
16  16 
2

So as  rotates clockwise around the ring Sk, G( )
rotates once around the origin in the counterclockwise
direction.Meanwhile, each pre-pole and pre-critical point
rotates clockwise by approximately 1/((n-2)nk-1 of a turn.

There are (n - 2)nk-1 pre-poles in the kth dynamical plane
ring, but (n - 2)nk-1 + 1 centers of Sierpinski holes in
the parameter plane rings. Here’s the reason:
1
1

On the annulus A, G( )  16  
16  16 
2

So as  rotates clockwise around the ring Sk, G( )
rotates once around the origin in the counterclockwise
direction.Meanwhile, each pre-pole and pre-critical point
rotates clockwise by approximately 1/((n-2)nk-1 of a turn.

So G( ) hits one additional prepole or pre-critical point
while traveling around each Sk.

Similar arguments show that each Sierpinski hole
on a Mandelpinski necklace is also surrounded
by infinitely many sub-Mandelpinski necklaces
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Similar arguments show that each Sierpinski hole
on a Mandelpinski necklace is also surrounded
by infinitely many sub-Mandelpinski necklaces
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Similar arguments show that each Sierpinski hole
on a Mandelpinski necklace is also surrounded
by infinitely many sub-Mandelpinski necklaces
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Similar arguments show that each Sierpinski hole
on a Mandelpinski necklace is also surrounded
by infinitely many sub-Mandelpinski necklaces
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Similar arguments show that each Sierpinski hole
on a Mandelpinski necklace is also surrounded
by infinitely many sub-Mandelpinski necklaces
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Some open problems:
Some open problems:
1. By Xiaoguang Wang’s result yesterday, the boundary
of B is always a simple closed curve (except when J
is a Cantor set) when n > 2.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
n=3
Some open problems:
1. By Xiaoguang Wang’s result yesterday, the boundary
of B is always a simple closed curve (except when J
is a Cantor set) when n > 2. Is the boundary of the
parameter plane also a simple closed curve???
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
n=3
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
n=3
Some open problems:
1. By Xiaoguang Wang’s result yesterday, the boundary
of B is always a simple closed curve (except when J
is a Cantor set) when n > 2. Is the boundary of the
parameter plane also a simple closed curve???
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
n=4
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
n=4
Some open problems:
1. By Xiaoguang Wang’s result yesterday, the boundary
of B is always a simple closed curve (except when J
is a Cantor set) when n > 2. Is the boundary of the
parameter plane also a simple closed curve???
2. What about the crazy case n = 2???
Some open problems:
1. By Xiaoguang Wang’s result yesterday, the boundary
of B is always a simple closed curve (except when J
is a Cantor set) when n > 2. Is the boundary of the
parameter plane also a simple closed curve???
2. What about the crazy case n = 2???
3. Are the Julia sets for these maps always locally connected?
Some open problems:
1. By Xiaoguang Wang’s result yesterday, the boundary
of B is always a simple closed curve (except when J
is a Cantor set) when n > 2. Is the boundary of the
parameter plane also a simple closed curve???
2. What about the crazy case n = 2???
3. Are the Julia sets for these maps always locally connected?
4. Are the parameter planes locally connected???
5. What is going on in the parameter plane near 0
when n = 2?
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
5. What is going on in the parameter plane near 0
when n = 2?
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
5. What is going on in the parameter plane near 0
when n = 2?
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
6. What is the structure in the parameter plane outside
the dividing circle?
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
7. What is going on in the parameter plane for the maps
F (z)  z 
n

z

QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
n = 2, d = 1
Not a baby
M-set
7. What is going on in the parameter plane for the maps
F (z)  z 
n

z

No Cantor
necklace
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
n = 2, d = 1
Not a baby
M-set
7. What is going on in the parameter plane for the maps
F (z)  z 
n


No Cantor
necklace
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
n = 2, d = 1
z
7. What is going on in the parameter plane for the maps
F (z)  z 
n


J approaches the
unit disk only along
these 3 lines
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
n = 4, d = 1
z