structures in the parameter planes dynamics of the family of complex maps paul blanchard toni garijo...
Post on 22-Dec-2015
212 views
TRANSCRIPT
![Page 1: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/1.jpg)
Structures in the Parameter Planes
Dynamics of the family of complex maps
Paul BlanchardToni GarijoMatt HolzerU. HoomiforgotDan LookSebastian Marotta Monica Moreno RochaElizabeth RussellYakov ShapiroDavid Uminsky
with:
€
Fλ (z) = z n +λ
z n
![Page 2: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/2.jpg)
First a brief advertisement:
AIMS Conference on Dynamical Systems, Differential Equations and Applications
Dresden University of TechnologyDresden, GermanyMay 25-28 2010
Organizers: Janina Kotus, Xavier Jarque, me
One half hour slots for speakers. Interested in attending/speaking?
Contact me at [email protected]
![Page 3: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/3.jpg)
![Page 4: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/4.jpg)
Structures in the Parameter Planes
What you see in the dynamical plane often reappears(enchantingly so) in the parameter plane....
Dynamics of the family of complex maps
€
Fλ (z) = z n +λ
z n
![Page 5: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/5.jpg)
Cantor Necklaces:
A Cantor necklace in aJulia set when n = 2
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 6: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/6.jpg)
Cantor Necklaces:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Cantor necklace in aJulia set when n = 2
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
and in the parameter plane
![Page 7: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/7.jpg)
Cantor Necklaces:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Cantor necklace in aJulia set when n = 2
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
and in the parameter plane
![Page 8: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/8.jpg)
Cantor Necklaces:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Cantor necklace in aJulia set when n = 2
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
and in the parameter plane
![Page 9: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/9.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Mandelpinski Necklaces:
Circles of preimages ofthe trap door and critical
points around 0
![Page 10: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/10.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Mandelpinski Necklaces:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Circles through centers ofSierpinski holes and baby M -sets in the param-plane
Circles of preimages ofthe trap door and critical
points around 0
![Page 11: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/11.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Mandelpinski Necklaces:
Circles of pre-preimages ofthe trap door and pre-critical
points around 0
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
* the only exception
Circles through centers ofSierpinski holes and baby M*-sets in the param-plane
![Page 12: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/12.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Mandelpinski Necklaces:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Circles of pre-preimages ofthe trap door and pre-critical
points around 0
* the only exception
Circles through centers ofSierpinski holes and baby M*-sets in the param-plane
![Page 13: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/13.jpg)
Mandelpinski Necklaces:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Circles of pre-preimages ofthe trap door and pre-critical
points around 0
Circles through centers ofSierpinski holes and baby M -sets in the param-plane
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 14: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/14.jpg)
Mandelpinski Necklaces:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Circles of pre-preimages ofthe trap door and pre-critical
points around 0
Circles through centers ofSierpinski holes and baby M -sets in the param-plane
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 15: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/15.jpg)
Mandelpinski Necklaces:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Circles of pre-preimages ofthe trap door and pre-critical
points around 0
Circles through centers ofSierpinski holes and baby M -sets in the param-plane
![Page 16: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/16.jpg)
Mandelpinski Necklaces:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Circles through Sierpinskiholes and baby Mandelbrotsets in the parameter plane
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Circles of pre-preimages ofthe trap door and pre-critical
points around 0
![Page 17: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/17.jpg)
As Douady often said “You sow the seeds in thedynamical plane and reap the harvest in the
parameter plane.”
It is often easy to prove something in the dynamicalplane, but harder to reproduce it in the parameter plane.
Here is how we will do this:
![Page 18: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/18.jpg)
Suppose you have some object in the dynamicalplane that varies analytically with the parametermaybe a closed curve, maybe a Cantor necklace, or:
€
λ
dynamical plane
![Page 19: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/19.jpg)
Suppose you have some object in the dynamicalplane that varies analytically with the parametermaybe a closed curve, maybe a Cantor necklace, or:
Maybe it’s your face,so call it Face( )
€
λ
dynamical plane
€
λ
![Page 20: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/20.jpg)
Suppose you have some object in the dynamicalplane that varies analytically with the parametermaybe a closed curve, maybe a Cantor necklace, or:
Maybe it’s your face,so call it Face( )Change , and Face( )moves analytically:
€
λ
€
λ
€
λ
dynamical plane
€
λ
![Page 21: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/21.jpg)
Suppose you have some object in the dynamicalplane that varies analytically with the parametermaybe 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
€
λ
€
λ
€
λ
dynamical plane
€
λ
![Page 22: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/22.jpg)
Suppose you have some object in the dynamicalplane that varies analytically with the parametermaybe 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 socute!)
€
λ
€
λ
€
λ
dynamical plane
€
λ
![Page 23: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/23.jpg)
So any particular point in Face( ), say the tip ofyour nose, nose( ), varies analytically with
€
λ
€
λ
€
λ
dynamical plane
nose( )
€
λ
![Page 24: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/24.jpg)
So any particular point in Face( ), say the tip ofyour nose, nose( ), varies analytically with
€
λ
€
λ
dynamical plane
nose( )
€
λ€
λ
![Page 25: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/25.jpg)
So any particular point in Face( ), say the tip ofyour nose, nose( ), varies analytically with
€
λ
€
λ
dynamical plane
nose( )
€
λ€
λ
![Page 26: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/26.jpg)
So we have an analytic function nose( ) fromparameter plane to the dynamical plane
€
λ
dynamical plane
nose( )
€
λ
parameter plane
![Page 27: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/27.jpg)
So we have an analytic function nose( ) fromparameter plane to the dynamical plane
€
λ
€
λ
dynamical plane
nose( )
€
λ
parameter plane
![Page 28: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/28.jpg)
Now suppose we have another analytic function G( ) taking parameter plane to dynamical plane one-to-one
€
λ
dynamical plane
nose( )
€
λ
parameter plane
€
λ
G
![Page 29: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/29.jpg)
So we have an inverse map G-1 taking the dynamicalplane back to the parameter plane
€
λ
dynamical plane
nose( )
€
λ
parameter plane
G-1
![Page 30: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/30.jpg)
Now suppose G takes a compact disk D in the parameterplane to a disk in dynamical plane, and nose( ) is alwayscontained strictly inside G(D) when .
€
λ
dynamical plane
nose( )
€
λ
parameter plane
G-1
€
λ
D G(D)
€
λ ∈D
![Page 31: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/31.jpg)
So G-1(nose( )) maps D strictly inside itself.
€
λ
dynamical plane
nose( )
€
λ
parameter plane
G-1
€
λ
D G(D)
![Page 32: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/32.jpg)
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( )).
€
λ*
dynamical planeparameter plane
G-1
€
λ
D G(D)
€
λ
€
λ*
nose( )
€
λ*
![Page 33: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/33.jpg)
is the unique parameter for which G( ) = nose( ).
€
λ*
dynamical planeparameter plane
G-1D G(D)
€
λ*
€
λ*
nose( )
€
λ*€
λ*
![Page 34: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/34.jpg)
is the unique parameter for which G( ) = nose( ).
dynamical planeparameter plane
DG(D)
€
λ*
€
λ*
€
λ*
If we do this for each point in Face( ), we then get the same “object” in the parameter plane.
€
λ
![Page 35: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/35.jpg)
is the unique parameter for which G( ) = nose( ).
dynamical planeparameter plane
DG(D)
€
λ*
€
λ*
€
λ*
If we do this for each point in Face( ), we then get the same “object” in the parameter plane.
€
λ
Why are you so unhappyliving in the parameter plane?
![Page 36: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/36.jpg)
The goal today is to show the existence in theparameter plane of:
1. Cantor necklaces 2. Cantor webs 3. Mandelpinski necklaces 4. Cantor sets of circles of Sierpinski curve Julia sets
![Page 37: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/37.jpg)
1. Cantor Necklaces
A Cantor necklace is the Cantor middle thirds set with open disks replacing the removed intervals.
![Page 38: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/38.jpg)
1. Cantor Necklaces
A Cantor necklace is the Cantor middle thirds set with open disks replacing the removed intervals.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
a Julia set with n = 2 anda Cantor necklace
![Page 39: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/39.jpg)
1. Cantor Necklaces
A Cantor necklace is the Cantor middle thirds set with open disks replacing the removed intervals.
a Julia set with n = 2 andanother Cantor necklace
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 40: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/40.jpg)
1. Cantor Necklaces
A Cantor necklace is the Cantor middle thirds set with open disks replacing the removed intervals.
a Julia set with n = 2 andlots of Cantor necklaces
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 41: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/41.jpg)
And there are Cantor necklaces in the parameter planes.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 2
1. Cantor Necklaces
![Page 42: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/42.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 2
1. Cantor Necklaces
![Page 43: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/43.jpg)
We’ll just show the existence of this Cantor necklacealong the negative real axis when n = 2.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 2
1. Cantor Necklaces
![Page 44: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/44.jpg)
€
Fλ (z ) = z 3 +λ
z 3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Recall:
B = immediate basin of T = trap door
€
∞
B
T
![Page 45: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/45.jpg)
2n free critical points
€
cλ = λ1/2n€
Fλ (z ) = z 3 +λ
z 3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
cc
Recall:
B = immediate basin of T = trap door
€
∞
B
T
![Page 46: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/46.jpg)
2n free critical points
€
cλ = λ1/2n
Only 2 critical values
€
vλ = ±2 λ
€
Fλ (z ) = z 3 +λ
z 3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
cc
v
Recall:
B = immediate basin of T = trap door
€
∞
B
T
![Page 47: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/47.jpg)
2n free critical points
€
cλ = λ1/2n
Only 2 critical values
€
vλ = ±2 λ
€
Fλ (z ) = z 3 +λ
z 3
2n prepoles
€
pλ = (−λ )1/2n
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
cc
v
p
p
Recall:
B = immediate basin of T = trap door
€
∞
B
0
![Page 48: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/48.jpg)
Consider
€
Fλ (z) = z2 +λ
z2 , λ < 0
![Page 49: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/49.jpg)
Since , preserves the real line.
Consider
€
Fλ (z) = z2 +λ
z2 , λ < 0
€
Fλ
€
λ < 0
![Page 50: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/50.jpg)
Consider
€
Fλ (z) = z2 +λ
z2 , λ < 0
graph of
€
Fλ (x)
need a glassof wine???
![Page 51: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/51.jpg)
Consider
€
Fλ (z) = z2 +λ
z2 , λ < 0
B = basin of infinity
graph of
€
Fλ (x)
![Page 52: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/52.jpg)
Consider
€
Fλ (z) = z2 +λ
z2 , λ < 0
B = basin of infinity
T = trap door
graph of
€
Fλ (x)
![Page 53: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/53.jpg)
Consider
€
Fλ (z) = z2 +λ
z2 , λ < 0
B = basin of infinity
T = trap door
graph of
€
Fλ (x)
I0
I1The two intervals I0 andI1 are expanded over theunion of these intervalsand the trap door.
![Page 54: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/54.jpg)
Consider
€
Fλ (z) = z2 +λ
z2 , λ < 0
B = basin of infinity
T = trap door
graph of
€
Fλ (x)
I0
So there is an invariant Cantorset on the negative real axis.
The two intervals I0 andI1 are expanded over theunion of these intervalsand the trap door.
I1
![Page 55: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/55.jpg)
Consider
€
Fλ (z) = z2 +λ
z2 , λ < 0
B = basin of infinity
T = trap door
graph of
€
Fλ (x)
I0
So there is an invariant Cantorset on the negative real axis.Add in the preimages of Tto get the Cantor necklace inthe dynamical plane for .
The two intervals I0 andI1 are expanded over theunion of these intervalsand the trap door.
I1
€
λ < 0
![Page 56: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/56.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
The Cantor necklace for negative
€
λ
![Page 57: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/57.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
This portion is also a Cantor necklacelying on the negative real axis for
€
λ < 0
![Page 58: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/58.jpg)
And we have a similar Cantor necklacelying on the negative real axis in theparameter plane for n = 2.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 59: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/59.jpg)
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
D
![Page 60: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/60.jpg)
Let be the second iterate of the critical point
€
G(λ ) = Fλ (vλ )
D
€
G(λ )
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
![Page 61: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/61.jpg)
Let be the second iterate of the critical point
€
G(λ ) = Fλ (vλ ) = (2 λ )2 +λ
(2 λ )2
D
€
G(λ )
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
![Page 62: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/62.jpg)
Let be the second iterate of the critical point
€
G(λ ) = Fλ (vλ ) = (2 λ )2 +λ
(2 λ )2= 4λ +
1
4
D
€
G(λ )
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
![Page 63: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/63.jpg)
Let be the second iterate of the critical point
So G is 1-to-1 on D,and maps D over itself;
D
G
€
G(λ )
€
G(λ ) = Fλ (vλ ) = (2 λ )2 +λ
(2 λ )2= 4λ +
1
4
.25-3.75
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
![Page 64: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/64.jpg)
Let be the second iterate of the critical point
So G is 1-to-1 on D,and maps D over itself;equivalently, G-1 contractsG(D) inside itself. D
G-1
€
G(λ )
€
G(λ ) = Fλ (vλ ) = (2 λ )2 +λ
(2 λ )2= 4λ +
1
4
.25-3.75
To see this, let D be the half-disk |z| < 1, Re(z) < 0.
![Page 65: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/65.jpg)
For any in D (not just ),we also have an invariant Cantor set as we showed earlier:
€
λ
€
λ < 0
![Page 66: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/66.jpg)
U2
U0
€
cλ
For any in D (not just ),we also have an invariant Cantor set as we showed earlier:
€
λ
€
λ < 0
U0 and U2 are portionsof a prepole sector
![Page 67: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/67.jpg)
U2
U0
€
cλ
For any in D (not just ),we also have an invariant Cantor set as we showed earlier:
€
λ
€
λ < 0
U0 and U2 are portionsof a prepole sectorthat are each mappedunivalently over bothU0 and U2.
![Page 68: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/68.jpg)
U0
€
cλ
U0 and U2 are portionsof a prepole sectorthat are each mappedunivalently over bothU0 and U2.
So there is a portion of a Cantor set lying in U2.
For any in D (not just ),we also have an invariant Cantor set as we showed earlier:
€
λ
€
λ < 0
![Page 69: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/69.jpg)
U0
For any in D (not just ),we also have an invariant Cantor set as we showed earlier:
€
λ
U0 and U2 are portionsof a prepole sectorthat are each mappedunivalently over bothU0 and U2.
€
cλ
So there is a portion of a Cantor set lying in U2.
And we can add in theappropriate preimages ofthe trap door to get aCantor necklace.
€
λ < 0
![Page 70: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/70.jpg)
And, since lies in D, theCantor set lies inside G(D).
€
λ
G(D)
For any in D (not just ),we also have an invariant Cantor set as we showed earlier:
€
λ
€
λ < 0
U0 and U2 are portionsof a prepole sectorthat are each mappedunivalently over bothU0 and U2.
So there is a portion of a Cantor set lying in U2.
And we can add in theappropriate preimages ofthe trap door to get aCantor necklace.
![Page 71: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/71.jpg)
U0
U2
We can identify each point in the Cantor set in U2 bya unique sequence of 0’s and 2’s: s = (2 s1 s2 s3 ....)given by the itinerary of the point.
![Page 72: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/72.jpg)
So, for each such sequence s,we have a map , which is defined on D anddepends analytically on
€
λ → zs (λ )
€
λ
We can identify each point in the Cantor set in U2 bya unique sequence of 0’s and 2’s: s = (2 s1 s2 s3 ....)given by the itinerary of the point.
U0
U2€
zs (λ )
![Page 73: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/73.jpg)
We therefore have two maps defined on D:
D
G(D)
![Page 74: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/74.jpg)
We therefore have two maps defined on D:
1. The univalent map
€
G(λ ) = 4λ +1
4
D
G G(D)
![Page 75: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/75.jpg)
We therefore have two maps defined on D:
1. The univalent map
€
G(λ ) = 4λ +1
4
D
G
2. The point in the Cantor set
€
zs (λ )
€
zs
G(D)
![Page 76: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/76.jpg)
D
G-1
€
G−1 ozs
€
zs
G(D)
So maps D strictly inside itself;
![Page 77: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/77.jpg)
D
G-1
€
G−1 ozs
€
zs
G(D)
So maps D strictly inside itself; bythe Schwarz Lemma, there is a unique fixed point in D for this map.
€
λs*
€
λs*
![Page 78: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/78.jpg)
For this parameter, we have , sothis is the unique parameter for which the criticalorbit lands on the point .
€
G(λ s* ) = zs (λ s
* )
D
G-1
€
G−1 ozs
€
zs
G(D)€
zs (λ )
So maps D strictly inside itself; bythe Schwarz Lemma, there is a unique fixed point in D for this map.
€
λs*
D
€
λs*
![Page 79: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/79.jpg)
Claim: this Cantor set lies on the negative real axis.
This produces a Cantor set of parameters ,one for each sequence s.
€
λs*
![Page 80: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/80.jpg)
€
G(λ ) = 4λ +1/ 4
Recall:
, so G decreases from .25 to -3.75 as goes from 0 to -1 in D.
€
λ
Claim: this Cantor set lies on the negative real axis.
This produces a Cantor set of parameters ,one for each sequence s.
€
λs*
![Page 81: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/81.jpg)
, so G decreases from .25 to -3.75 as goes from 0 to -1 in D.
Recall:
Claim: this Cantor set lies on the negative real axis.
the Cantor set in the dynamical planelies on the negative real axis when .
€
λ < 0
This produces a Cantor set of parameters ,one for each sequence s.
€
λs*
€
G(λ ) = 4λ +1/ 4
€
λ
![Page 82: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/82.jpg)
Recall:
So must hit each point in the Cantor set alongthe negative axis at least once.
€
G(λ )
Claim: this Cantor set lies on the negative real axis.
the Cantor set in the dynamical planelies on the negative real axis when .
€
λ < 0
This produces a Cantor set of parameters ,one for each sequence s.
€
λs*
€
G(λ ) = 4λ +1/ 4 , so G decreases from .25 to -3.75 as goes from 0 to -1 in D.
€
λ
![Page 83: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/83.jpg)
Recall:
So must hit each point in the Cantor set alongthe negative axis at least once.
€
G(λ )
So each parameter in the parameter plane necklacemust also lie in [-1, 0]. This produces the Cantor set portion of the necklace on the negative real axis.
€
λs*
Claim: this Cantor set lies on the negative real axis.
the Cantor set in the dynamical planelies on the negative real axis when .
€
λ < 0
This produces a Cantor set of parameters ,one for each sequence s.
€
λs*
€
G(λ ) = 4λ +1/ 4 , so G decreases from .25 to -3.75 as goes from 0 to -1 in D.
€
λ
![Page 84: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/84.jpg)
Similar arguments produce parameters onthe negative axis that land after a specifieditinerary on a particular point in B (thatis determined by the Böttcher coordinate).
![Page 85: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/85.jpg)
Similar arguments produce parameters onthe negative axis that land after a specifieditinerary on a particular point in B (thatis determined by the Böttcher coordinate).And then these intervals can be expandedto get the Sierpinski holes in the necklace.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 86: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/86.jpg)
2. Cantor webs
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 4
Recall that, when n > 2, we have Cantor “webs” in the dynamical plane:
n = 3
![Page 87: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/87.jpg)
2. Cantor webs
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 4
Recall that, when n > 2, we have Cantor “webs” in the dynamical plane:
n = 3
![Page 88: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/88.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2. Cantor webs
Recall that, when n > 2, we have Cantor “webs” in the dynamical plane:
n = 3n = 3
![Page 89: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/89.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2. Cantor webs
When n > 2, we also have Cantor “webs” in the parameter plane:
n = 4n = 3
![Page 90: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/90.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
2. Cantor webs
When n > 2, we also have Cantor “webs” in the parameter plane:
n = 4n = 3
![Page 91: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/91.jpg)
2. Cantor webs
When n > 2, we also have Cantor “webs” in the parameter plane:
n = 4n = 3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 92: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/92.jpg)
A slightly different argument as in the case of theCantor necklaces works here. Say n = 3.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 3
U1
U2
U4
U5€
vλ
€
−vλ
In the dynamical plane, wehad the disks Uj.
![Page 93: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/93.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 3
U1
U2
U4
U5€
vλ
€
−vλ
Each of these Uj were mappedunivalently over all the others,
excluding U0 and Un, sowe found an invariant Cantor
set in these regions.
In the dynamical plane, wehad the disks Uj.
U0
U3
A slightly different argument as in the case of theCantor necklaces works here. Say n = 3.
![Page 94: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/94.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 3
U1
U2
U4
U5€
vλ
€
−vλ
Each of these Uj were mappedunivalently over all the others,
excluding U0 and U3, sowe found an invariant Cantor
set in these regions.
In the dynamical plane, wehad the disks Uj.
U0
U3
U0 and U3 are mappedunivalently over these Uj,
so there is a preimage ofthis Cantor set in both U0
and U3
A slightly different argument as in the case of theCantor necklaces works here. Say n = 3.
![Page 95: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/95.jpg)
Now let be one of the two critical values, so
€
G(λ )
€
G(λ ) = 2 λ
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
U2
U4
€
−vλ
U3
D
And choose a disk D in one of the “symmetry sectors”
in the parameter plane:
![Page 96: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/96.jpg)
Now let be one of the two critical values, so
€
G(λ )
€
G(λ ) = 2 λ
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
U1
U2
U4
U5€
vλ
€
−vλ
U0
U3
D
And choose a disk D in one of the “symmetry sectors”
in the parameter plane:
Then G maps D univalentlyover all of U0, so we again get a copy of the Cantor set in D
G
![Page 97: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/97.jpg)
Now let be one of the two critical values, so
€
G(λ )
€
G(λ ) = 2 λ
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
U1
U2
U4
U5€
vλ
€
−vλ
U0
U3
D
And choose a disk D in one of the “symmetry sectors”
in the parameter plane:
Then G maps D univalentlyover all of U0, so we again get a copy of the Cantor set in D
G
Then adjoining the appropriate Sierpinski holesgives a Cantor web in the parameter plane.
![Page 98: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/98.jpg)
3. “Mandelpinski” necklaces
A Mandlepinski necklace is a simple closed curve inthe parameter plane that passes alternately through acertain number of centers of baby M-sets and thesame number of centers of S-holes.
![Page 99: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/99.jpg)
oops, sorry....
A Mandlepinski necklace is a simple closed curve inthe parameter plane that passes alternately through acertain number of centers of baby M-sets and thesame number of centers of S-holes.
3. “Mandelpinski” necklaces
![Page 100: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/100.jpg)
A Mandlepinski necklace is a simple closed curve inthe parameter plane that passes alternately through acertain number of centers of baby M-sets and thesame number of centers of Sierpinski-holes.
3. “Mandelpinski” necklaces
![Page 101: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/101.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Julia set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
parameter plane n = 4
3. “Mandelpinski” necklaces
![Page 102: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/102.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
There is a “ring” around Tpassing through 8 = 2*4
preimages of T
parameter plane n = 4
3. “Mandelpinski” necklaces
![Page 103: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/103.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
parameter plane n = 4 There is a “ring” around Tpassing through 8 = 2*4
preimages of T
3. “Mandelpinski” necklaces
![Page 104: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/104.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Another “ring” around Tpassing through 32 = 2*42
preimages of T
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
parameter plane n = 4
3. “Mandelpinski” necklaces
![Page 105: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/105.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Another “ring” around Tpassing through 32 = 2*42
preimages of T
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
parameter plane n = 4
3. “Mandelpinski” necklaces
![Page 106: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/106.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Another “ring” around Tpassing through 128 = 2*43
preimages of T
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
3. “Mandelpinski” necklaces
parameter plane n = 4
![Page 107: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/107.jpg)
parameter plane for n = 4
Now look around the McMullen domain in the parameter plane:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 108: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/108.jpg)
Now look around the McMullen domain in the parameter plane:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
There is a ring around M that passes alternately through the centers of
3 = 2*40 + 1 Sierpinski holes and 3 Mandelbrot sets
![Page 109: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/109.jpg)
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:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 110: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/110.jpg)
Another ring around M that passes alternately through the centers of
9 = 2*41 + 1 Sierpinski holes and 9 “Mandelbrot sets”*
Now look around the McMullen domain in the parameter plane:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
*well, 3 period 2 bulbs
![Page 111: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/111.jpg)
Another ring around M that passes alternately through the centers of
9 = 2*41 + 1 Sierpinski holes and 9 “Mandelbrot sets”*
Now look around the McMullen domain in the parameter plane:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
*well, 3 period 2 bulbs
![Page 112: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/112.jpg)
Then 33 = 2*42 + 1 Sierpinski holes and 33 Mandelbrot sets
Now look around the McMullen domain in the parameter plane:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 113: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/113.jpg)
Then 129 = 2*43 + 1 Sierpinski holes and 129 Mandelbrot sets
Now look around the McMullen domain in the parameter plane:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 114: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/114.jpg)
parameter plane for n = 3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Similar kinds of rings occur in the other parameter planes:
n = 3
![Page 115: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/115.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Similar kinds of rings occur in the other parameter planes:
S0: 2 = 1*30 + 1 Sierpinski holes & M-sets
S0
n = 3
![Page 116: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/116.jpg)
S1: 4 = 1*31 + 1 Sierpinski holes & M-sets*
*well, two period 2 bulbs
n = 3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 117: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/117.jpg)
S2: 10 = 1*32 + 1 Sierpinski holes & “M-sets”
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 3
![Page 118: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/118.jpg)
S3: 28 = 1*33 + 1 Sierpinski holes & M-sets
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 3
![Page 119: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/119.jpg)
82, 244, then 730 Sierpinski holes...
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 3
![Page 120: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/120.jpg)
the 13th ring passes through1,594,324 Sierpinski holes...
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 3
sorry, I forgot.....nevermind
![Page 121: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/121.jpg)
* with one exception
Theorem: For each n > 2, the McMullen domain issurrounded 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;
• Sk meets the center of exactly (n-2)nk-1 + 1 Sierpinski holes, each with escape time k + 2;1. Sk also passes through the centers of the same number of baby Mandelbrot sets*
![Page 122: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/122.jpg)
The critical points andprepoles all lie on the “critical circle”
€
r = | λ |1/2n pc
p
c
![Page 123: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/123.jpg)
The critical points andprepoles all lie on the “critical circle”
The critical circle is mapped 2n-to-1 ontothe “critical value ray”
v
0
pc
p
c
€
r = | λ |1/2n
![Page 124: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/124.jpg)
The critical points andprepoles all lie on the “critical circle”
The critical circle is mapped 2n-to-1 ontothe “critical value ray”
v
0
And every other circle centeredat the origin and outside thecritical circle is mapped n-to-1to an ellipse with foci at the criticalvalues
€
r = | λ |1/2n
![Page 125: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/125.jpg)
The critical points andprepoles all lie on the “critical circle”
The critical circle is mapped 2n-to-1 ontothe “critical value ray”
v
0
And every other circle centeredat the origin and outside thecritical circle is mapped n-to-1to an ellipse with foci at the criticalvalues
€
r = | λ |1/2n
![Page 126: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/126.jpg)
The critical points andprepoles all lie on the “critical circle”
The critical circle is mapped 2n-to-1 ontothe “critical value ray”
v
0
And every other circle centeredat the origin and outside thecritical circle is mapped n-to-1to an ellipse with foci at the criticalvalues, and same inside
€
r = | λ |1/2n
![Page 127: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/127.jpg)
There are no critical pointsoutside the critical circle, so this region is mapped asn-to-1 covering onto thecomplement of thecritical value ray.
v
0
![Page 128: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/128.jpg)
v
0 The interior of thecritical circle is also mapped n-to-1 onto thecomplement of thecritical value ray
There are no critical pointsoutside the critical circle, so this region is mapped asn-to-1 covering onto thecomplement of thecritical value ray.
![Page 129: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/129.jpg)
The dividing circle contains all parameters for whichthe critical values lie on the critical circle, i.e.,
€
| λ |1/2n= 2 | λ |1/2 ⇒ | λ | = 2−2n /(n−1)
![Page 130: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/130.jpg)
The dividing circle contains all parameters for whichthe critical values lie on the critical circle, i.e.,
€
| λ |1/2n= 2 | λ |1/2 ⇒ | λ | = 2−2n /(n−1)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
When n = 4, the dividing circle passes through 3
centers of Sierpinski holes and 3 baby Mandelbrot sets
€
n = 4 : | λ | = 2−8/3
![Page 131: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/131.jpg)
The dividing circle passes through n-1 centers of Sierpinski holes and n-1 centers of baby Mandelbrot sets.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
When n = 4, the dividing circle passes through 3
centers of Sierpinski holes and 3 baby Mandelbrot sets
€
n = 4 : | λ | = 2−8/3
![Page 132: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/132.jpg)
Reason:
€
r = | λ |8€
| λ | = 2−8/3
parameter plane n = 4 €
vλ
€
λ
dynamical plane
![Page 133: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/133.jpg)
Reason: as runs once around the dividing circle,
€
r = | λ |8€
| λ | = 2−8/3
€
λ
parameter plane n = 4 €
vλ
€
λ
dynamical plane
![Page 134: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/134.jpg)
€
r = | λ |8
€
vλ
€
| λ | = 2−8/3€
λ
Reason: as runs once around the dividing circle, rotates 1/2 of a turn,
€
λ
€
vλ
parameter plane n = 4
dynamical plane
![Page 135: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/135.jpg)
Reason: as runs once around the dividing circle, rotates 1/2 of a turn, while the critical points and prepoles
each rotate on 1/8 of a turn.
€
r = | λ |8
€
λ
€
vλ
€
| λ | = 2−8/3€
λ
parameter plane n = 4
dynamical plane
€
vλ
![Page 136: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/136.jpg)
Reason: as runs once around the dividing circle, rotates 1/2 of a turn, while the critical points and prepoles
each rotate on 1/8 of a turn.
€
r = | λ |8
€
λ
€
vλ
€
vλ
So meets 3 prepoles and critical points enroute.
€
vλ
€
| λ | = 2−8/3€
λ
parameter plane n = 4
dynamical plane
![Page 137: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/137.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
So the ring S0 is just the dividing circle in parameter plane.
S0
n = 4
![Page 138: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/138.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
So the ring S0 is just the dividing circle in parameter plane.
S0
For the other rings, let’s consider for simplicity only the case where n = 4
n = 4
![Page 139: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/139.jpg)
When lies inside the dividing circle, we have
€
λ
€
| vλ | = 2 | | λ | < | λ |1/8 = | cλ |
![Page 140: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/140.jpg)
When lies inside the dividing circle, we have
€
λ
so maps the critical circle C0 strictly inside itself
€
Fλ
€
vλ
€
−vλ C0
€
| vλ | = 2 | | λ | < | λ |1/8 = | cλ |
![Page 141: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/141.jpg)
€
vλ
€
−vλ
Now there is a preimage C1 of the critical circle thatis mapped 4-to-1 onto the critical circle, and this curvecontains 32 pre-critical points and 32 pre-pre-poles.
C0
C1
![Page 142: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/142.jpg)
€
vλ
€
−vλ
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.
C0
C1
C2
![Page 143: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/143.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
The rings C0 and C1
![Page 144: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/144.jpg)
Let be the second iterate of the critical point
€
G(λ ) = Fλ2 (cλ ) = 2nλn /2 +
1
2nλ (n /2)−1
€
G(λ )
![Page 145: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/145.jpg)
Let be the second iterate of the critical point
€
G(λ ) = 16λ2 +1
16λSo when n = 4.
€
G(λ ) = Fλ2 (cλ ) = 2nλn /2 +
1
2nλ (n /2)−1
€
G(λ )
![Page 146: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/146.jpg)
Let be the second iterate of the critical point
€
G(λ ) = 16λ2 +1
16λSo when n = 4.
€
G(λ ) = Fλ2 (cλ ) = 2nλn /2 +
1
2nλ (n /2)−1
€
G(λ )
Note that
€
G(λ ) → ∞ as
€
λ → 0 provided n > 2.
When n = 2,
€
G(λ ) = 4λ +1
4, a very different situation.
![Page 147: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/147.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
G maps points in the parameter plane to points in the dynamical plane
C0
G
the critical circle
![Page 148: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/148.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
G
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)
C0
D
![Page 149: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/149.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
and G(D) covers C1, C2,...
C0
D
G
![Page 150: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/150.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Choose a small disk D0 inside M. Then G maps the annulus A = D - D0 univalently over all of the Cj, j > 0.
C0
D0
A
G
![Page 151: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/151.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Choose a parametrization of Ck, say . So wehave a second map from A into G(A),
C0
D0
A
€
λ → γλk (θ )
€
γλk (θ )
€
γλk (θ )
G
![Page 152: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/152.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Since G is 1-to-1, we thus have a map which takes A into A.
C0
D0
A
€
H (λ ) =G−1(γ λk (θ ))
€
γλk (θ )
H
![Page 153: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/153.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
C0
D0
A
€
γλk (θ )
H
Let S be the covering strip of A and let H*: S S be the covering map of H: A A
![Page 154: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/154.jpg)
Let S be the covering strip of A and let H*: S S be the covering map of H: A A
By the Schwarz Lemma, for each given k, , and , there is a unique fixed point for H* in A, which depends analytically on .
€
ηλk (θ )
€
θ
€
λ
€
λ
![Page 155: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/155.jpg)
Let S be the covering strip of A and let H*: S S be the covering map of H: A A
By the Schwarz Lemma, for each given k, , and , there is a unique fixed point for H* in A, which depends analytically on .
€
ηλk (θ )
€
θ
€
λ
So the map gives a parametrization of thering Sk in the parameter plane, and -values that
correspond to pre-poles or pre-critical points are thenparameters at the centers of Sierpinski holes or baby
Mandelbrot sets.
€
λ
€
λ → ηλk (θ )
€
θ
![Page 156: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/156.jpg)
There are (n - 2)nk-1 pre-poles in the kth dynamical planering, but (n - 2)nk-1 + 1 centers of Sierpinski holes inthe parameter plane rings. Here’s the reason:
![Page 157: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/157.jpg)
On the annulus A,
€
G(λ ) = 16λ2 +1
16λ≈
1
16λ
There are (n - 2)nk-1 pre-poles in the kth dynamical planering, but (n - 2)nk-1 + 1 centers of Sierpinski holes inthe parameter plane rings. Here’s the reason:
![Page 158: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/158.jpg)
On the annulus A,
€
G(λ ) = 16λ2 +1
16λ≈
1
16λ
So as rotates clockwise around the ring Sk, rotates once around the origin in the counterclockwise direction. Meanwhile, each pre-pole and pre-critical pointrotates clockwise by approximately 1/((n-2)nk-1 of a turn.
€
G(λ )
€
λ
There are (n - 2)nk-1 pre-poles in the kth dynamical planering, but (n - 2)nk-1 + 1 centers of Sierpinski holes inthe parameter plane rings. Here’s the reason:
![Page 159: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/159.jpg)
On the annulus A,
€
G(λ ) = 16λ2 +1
16λ≈
1
16λ
So hits one additional prepole or pre-critical pointwhile traveling around each Sk.
€
G(λ )
There are (n - 2)nk-1 pre-poles in the kth dynamical planering, but (n - 2)nk-1 + 1 centers of Sierpinski holes inthe parameter plane rings. Here’s the reason:
So as rotates clockwise around the ring Sk, rotates once around the origin in the counterclockwise direction. Meanwhile, each pre-pole and pre-critical pointrotates clockwise by approximately 1/((n-2)nk-1 of a turn.
€
G(λ )
€
λ
![Page 160: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/160.jpg)
Similar arguments show that each Sierpinski holeon a Mandelpinski necklace is also surroundedby infinitely many sub-Mandelpinski necklaces
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 161: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/161.jpg)
Similar arguments show that each Sierpinski holeon a Mandelpinski necklace is also surroundedby infinitely many sub-Mandelpinski necklaces
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 162: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/162.jpg)
Similar arguments show that each Sierpinski holeon a Mandelpinski necklace is also surroundedby infinitely many sub-Mandelpinski necklaces
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 163: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/163.jpg)
Similar arguments show that each Sierpinski holeon a Mandelpinski necklace is also surroundedby infinitely many sub-Mandelpinski necklaces
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 164: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/164.jpg)
Similar arguments show that each Sierpinski holeon a Mandelpinski necklace is also surroundedby infinitely many sub-Mandelpinski necklaces
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 165: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/165.jpg)
Some open problems:
![Page 166: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/166.jpg)
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 aTIFF (LZW) decompressor
are needed to see this picture.
n = 3
![Page 167: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/167.jpg)
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 aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 3 n = 3
![Page 168: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/168.jpg)
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???
n = 4 n = 4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 169: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/169.jpg)
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???
![Page 170: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/170.jpg)
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?
![Page 171: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/171.jpg)
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???
![Page 172: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/172.jpg)
5. What is going on in the parameter plane near 0 when n = 2?
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 173: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/173.jpg)
5. What is going on in the parameter plane near 0 when n = 2?
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 174: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/174.jpg)
5. What is going on in the parameter plane near 0 when n = 2?
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 175: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/175.jpg)
6. What is the structure in the parameter plane outside the dividing circle?
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 176: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/176.jpg)
7. What is going on in the parameter plane for the maps
€
Fλ (z) = zn +λ
z
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 2, d = 1
Not a babyM-set
![Page 177: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/177.jpg)
€
Fλ (z) = zn +λ
z
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 2, d = 1
No Cantornecklace
7. What is going on in the parameter plane for the maps
Not a babyM-set
![Page 178: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/178.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Fλ (z) = zn +λ
z
n = 2, d = 1
No Cantornecklace
7. What is going on in the parameter plane for the maps
![Page 179: Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta](https://reader038.vdocument.in/reader038/viewer/2022110323/56649d7f5503460f94a62137/html5/thumbnails/179.jpg)
€
Fλ (z) = zn +λ
z
n = 4, d = 1
7. What is going on in the parameter plane for the maps
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
J approaches theunit disk only alongthese 3 lines