topological structures in the julia sets of rational maps dynamics of the family of complex maps...
DESCRIPTION
Three different types of topological objects: 1. Cantor Necklaces dynamical plane n = 2TRANSCRIPT
![Page 1: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/1.jpg)
Topological Structures in the Julia Setsof Rational Maps
Dynamics of the family of complex maps
Paul Blanchard Mark MorabitoToni Garijo Monica Moreno RochaMatt Holzer Kevin PilgrimU. Hoomiforgot Elizabeth RussellDan Look Yakov ShapiroSebastian Marotta David Uminsky
with:
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Fλ (z) = z n + λz n
![Page 2: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/2.jpg)
Three different types of topological objects:
1. Cantor Necklaces
A Cantor necklace is a planar set that ishomeomorphic to the Cantor middle thirds
set with open disks replacing removed intervals.
![Page 3: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/3.jpg)
Three different types of topological objects:
1. Cantor Necklaces
QuickTime™ and a decompressor
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dynamical planen = 2
![Page 4: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/4.jpg)
Three different types of topological objects:
1. Cantor Necklaces
QuickTime™ and a decompressor
are needed to see this picture.
dynamical planen = 2
![Page 5: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/5.jpg)
Three different types of topological objects:
1. Cantor Necklaces
QuickTime™ and a decompressor
are needed to see this picture.
dynamical planen = 2
![Page 6: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/6.jpg)
Three different types of topological objects:
1. Cantor Necklaces
QuickTime™ and a decompressor
are needed to see this picture.QuickTime™ and a
decompressorare needed to see this picture.
dynamical plane parameter plane
n = 2
![Page 7: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/7.jpg)
Three different types of topological objects:
2. Mandelpinski Necklaces
Infinitely many simple closed curves in the parameter planethat pass alternately through centers of “Sierpinski holes”
and centers of baby Mandelbrot sets.
![Page 8: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/8.jpg)
Three different types of topological objects:
2. Mandelpinski Necklaces
parameter plane zoom inn = 3
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Three different types of topological objects:
3. CanManPinski Trees
A tree of Cantor necklaces with Mandelbrot sets interspersed between each branch
![Page 10: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/10.jpg)
Three different types of topological objects:
3. CanManPinski Trees
parameter plane parameter plane
n = 2
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QuickTime™ and a decompressor
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![Page 11: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/11.jpg)
Three different types of topological objects:
3. CanManPinski Trees
parameter plane parameter plane
n = 2
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QuickTime™ and a decompressor
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Dynamics of
complex and
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J ( Fλ
)The Julia set is:The closure of the set of repelling periodic points;The boundary of the escaping orbits;The chaotic set.
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Fλ (z) = z n + λz n
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λ,z
€
n ≥ 2
The Fatou set is the complement of .
€
J ( Fλ
)
![Page 13: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/13.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
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F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
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λ = 0
![Page 14: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/14.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 15: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/15.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 16: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/16.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 17: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/17.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 18: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/18.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 19: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/19.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 20: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/20.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 21: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/21.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 22: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/22.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 23: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/23.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 24: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/24.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 25: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/25.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 26: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/26.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 27: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/27.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 28: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/28.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 29: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/29.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 30: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/30.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 31: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/31.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 32: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/32.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 33: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/33.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 34: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/34.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 35: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/35.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 36: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/36.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 37: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/37.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 38: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/38.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 39: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/39.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 40: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/40.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 41: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/41.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 42: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/42.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 43: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/43.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 44: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/44.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 45: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/45.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 46: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/46.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 47: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/47.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 48: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/48.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 49: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/49.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 50: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/50.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 51: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/51.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 52: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/52.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 53: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/53.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 54: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/54.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 55: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/55.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 56: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/56.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 57: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/57.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 58: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/58.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 59: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/59.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
When , the Julia set is the unit circle
€
λ = 0
![Page 60: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/60.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
€
λ ≠ 0
€
λ = −1 / 16
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
But when , theJulia set explodes
When , the Julia set is the unit circle
€
λ = 0
![Page 61: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/61.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
€
λ ≠ 0
€
λ = −1 / 16
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
But when , theJulia set explodes
A Sierpinski curve
When , the Julia set is the unit circle
€
λ = 0
![Page 62: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/62.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
€
λ ≠ 0But when , theJulia set explodes
€
λ = −0 . 01
Another Sierpinski curve
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
When , the Julia set is the unit circle
€
λ = 0
![Page 63: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/63.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ = 0
€
F λ ( z ) = z2
+λ
z2
€
λ ≠ 0But when , theJulia set explodes
€
λ = −0 . 2
Also a Sierpinski curve
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
When , the Julia set is the unit circle
€
λ = 0
![Page 64: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/64.jpg)
A Sierpinski curve is any planar set that is homeomorphic to the Sierpinski carpet fractal.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
The Sierpinski Carpet
Sierpinski Curve
![Page 65: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/65.jpg)
Easy computations:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Fλ (z ) = z 3 +λz 3
€
λ=.036+.026i
2n free critical points
€
cλ = λ1/2n
![Page 66: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/66.jpg)
Easy computations:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ=.036+.026i
2n free critical points
€
cλ = λ1/2n€
Fλ (z ) = z 3 +λz 3
![Page 67: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/67.jpg)
Easy computations:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ=.036+.026i
2n free critical points
€
cλ = λ1/2n
Only 2 critical values
€
vλ = ±2 λ
€
Fλ (z ) = z 3 +λz 3
![Page 68: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/68.jpg)
Easy computations:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ=.036+.026i
2n free critical points
€
cλ = λ1/2n
Only 2 critical values
€
vλ = ±2 λ
€
Fλ (z ) = z 3 +λz 3
![Page 69: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/69.jpg)
Easy computations:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ=.036+.026i
2n free critical points
€
cλ = λ1/2n
Only 2 critical values
€
vλ = ±2 λ
€
Fλ (z ) = z 3 +λz 3
![Page 70: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/70.jpg)
Easy computations:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ=.036+.026i
2n free critical points
€
cλ = λ1/2n
Only 2 critical values
€
vλ = ±2 λ
But really only 1 freecritical orbit since
the map has 2n-foldsymmetry
€
Fλ (z ) = z 3 +λz 3
![Page 71: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/71.jpg)
Easy computations:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ=.036+.026i
is superattracting, so have immediate basin Bmapped n-to-1 to itself.
€
∞ B€
Fλ (z ) = z 3 +λz 3
![Page 72: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/72.jpg)
Easy computations:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ=.036+.026i
is superattracting, so have immediate basin Bmapped n-to-1 to itself.
B
T
€
Fλ (z ) = z 3 +λz 3
€
∞
0 is a pole, so havetrap door T mapped
n-to-1 to B.
![Page 73: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/73.jpg)
Easy computations:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ=.036+.026i
is superattracting, so have immediate basin Bmapped n-to-1 to itself.
B
0 is a pole, so havetrap door T mapped
n-to-1 to B.
T
€
Fλ (z ) = z 3 +λz 3
€
∞
So any orbit that eventuallyenters B must do so by
passing through T.
![Page 74: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/74.jpg)
The Escape Trichotomy
€
v λ ∈
€
J ( Fλ
)
€
⇒B
€
⇒
€
⇒
is a Cantor set
T
€
v λ ∈ is a Cantor set ofsimple closed curves
€
J ( Fλ
)
€
Fλk(v λ ) ∈ T
€
J ( Fλ
) is a Sierpinski curve
There are three distinct ways the critical orbit can enter B:
(this case does not occur if n = 2)
(with Dan Look & David Uminsky)
![Page 75: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/75.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
Case 1:
![Page 76: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/76.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
€
λ
![Page 77: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/77.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
€
λ
![Page 78: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/78.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ
![Page 79: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/79.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
€
λ
![Page 80: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/80.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
€
λ
![Page 81: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/81.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
€
λ
![Page 82: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/82.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
€
λ
![Page 83: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/83.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
€
λ
![Page 84: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/84.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
€
λ
![Page 85: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/85.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
€
λ
![Page 86: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/86.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
€
λ
![Page 87: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/87.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
€
v λ ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
parameter planewhen n = 3
J is a Cantor set
€
λ
![Page 88: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/88.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
parameter planewhen n = 3
Case 2: the critical values lie in T, not B
![Page 89: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/89.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
v λ ∈
€
⇒T
parameter planewhen n = 3
€
λ lies in the McMullen domain
€
λ
![Page 90: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/90.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
v λ ∈
€
⇒T
parameter planewhen n = 3
J is a Cantor set of simple closed curves
€
λ lies in the McMullen domain
€
λQuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
Remark: There is no McMullen domain in the case n = 2.
![Page 91: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/91.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
v λ ∈
€
⇒T
parameter planewhen n = 3
J is a Cantor set of simple closed curves
€
λ lies in the McMullen domain
€
λQuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
![Page 92: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/92.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
v λ ∈
€
⇒T
parameter planewhen n = 3
J is a Cantor set of simple closed curves
€
λ lies in the McMullen domain
€
λQuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
![Page 93: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/93.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
v λ ∈
€
⇒T
parameter planewhen n = 3
J is a Cantor set of simple closed curves
€
λ lies in the McMullen domain
€
λQuickTime™ and a
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€
v λ ∈
€
⇒T
parameter planewhen n = 3
J is a Cantor set of simple closed curves
€
λ lies in the McMullen domain
€
λQuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
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QuickTime™ and aTIFF (LZW) decompressor
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€
v λ ∈
€
⇒T
parameter planewhen n = 3
J is a Cantor set of simple closed curves
€
λ lies in the McMullen domain
€
λQuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
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QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
v λ ∈
€
⇒T
parameter planewhen n = 3
J is a Cantor set of simple closed curves
€
λ lies in the McMullen domain
€
λQuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
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€
⇒T
parameter planewhen n = 3
€
λ lies in a Sierpinski hole
€
Fλk(v λ ) ∈
Case 3: the critical orbit eventually lands in the trap door.
![Page 98: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/98.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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€
⇒T
parameter planewhen n = 3
J is a Sierpinski curve
€
λ lies in a Sierpinski hole
€
λ
€
Fλk(v λ ) ∈
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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QuickTime™ and aTIFF (LZW) decompressor
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€
⇒T
parameter planewhen n = 3
J is a Sierpinski curve
€
λ lies in a Sierpinski hole
€
λ
€
Fλk(v λ ) ∈
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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QuickTime™ and aTIFF (LZW) decompressor
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€
⇒T
parameter planewhen n = 3
J is a Sierpinski curve
€
λ lies in a Sierpinski hole
€
λ
€
Fλk(v λ ) ∈
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are needed to see this picture.
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QuickTime™ and aTIFF (LZW) decompressor
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€
⇒T
parameter planewhen n = 3
J is a Sierpinski curve
€
λ lies in a Sierpinski hole
€
λ
€
Fλk(v λ ) ∈
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QuickTime™ and aTIFF (LZW) decompressor
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€
⇒T
parameter planewhen n = 3
J is a Sierpinski curve
€
λ lies in a Sierpinski hole
€
λ
€
Fλk(v λ ) ∈
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QuickTime™ and aTIFF (LZW) decompressor
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€
⇒T
parameter planewhen n = 3
J is a Sierpinski curve
€
λ lies in a Sierpinski hole
€
λ
€
Fλk(v λ ) ∈
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QuickTime™ and aTIFF (LZW) decompressor
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€
⇒T
parameter planewhen n = 3
J is a Sierpinski curve
€
λ lies in a Sierpinski hole
€
λ
€
Fλk(v λ ) ∈
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€
⇒T
parameter planewhen n = 3
J is a Sierpinski curve
€
λ lies in a Sierpinski hole
€
λ
€
Fλk(v λ ) ∈
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QuickTime™ and aTIFF (LZW) decompressor
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⇒T
parameter planewhen n = 3
J is a Sierpinski curve
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λ lies in a Sierpinski hole
€
λ
€
Fλk(v λ ) ∈
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€
⇒T
parameter planewhen n = 3
J is a Sierpinski curve
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λ lies in a Sierpinski hole
€
λ
€
Fλk(v λ ) ∈
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⇒T
parameter planewhen n = 3
J is a Sierpinski curve
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λ lies in a Sierpinski hole
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λ
€
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1. Cantor necklaces in thedynamical and parameter plane
The Cantor necklace is homeomorphic to the Cantor middle thirds set with
open disks replacing removed intervals.
![Page 110: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/110.jpg)
1. Cantor necklaces in thedynamical and parameter plane
Julia set n = 2λ = -0.23
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The Cantor necklace is homeomorphic to the Cantor middle thirds set with
open disks replacing removed intervals.
![Page 111: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/111.jpg)
1. Cantor necklaces in thedynamical and parameter plane
parameter plane n = 4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
The Cantor necklace is homeomorphic to the Cantor middle thirds set with
open disks replacing removed intervals.
![Page 112: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/112.jpg)
Dynamical plane: n = 2
Suppose B and T are disjoint.
B
T
![Page 113: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/113.jpg)
Dynamical plane: n = 2
Four critical points
λ1/4
![Page 114: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/114.jpg)
Dynamical plane: n = 2
And two critical valuesthat do not lie in T
2λ1/2
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Dynamical plane: n = 2
The critical lines...
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Dynamical plane: n = 2
are mapped two-to-one toone of two critical value rays
![Page 117: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/117.jpg)
So the sectors S0 and S1 are mappedone-to-one to C - {critical value rays)
S1 S0
![Page 118: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/118.jpg)
And the regions I0 - T and I1 - T are mapped
one-to-one to C - B - {critical value rays)
Dynamical plane: n = 2
I1 I0
![Page 119: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/119.jpg)
And the regions I0 - T and I1 - T are mapped
one-to-one to C - B - {critical value rays)
Dynamical plane: n = 2
![Page 120: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/120.jpg)
Dynamical plane: n = 2
I1 I0
So consider the bow-tie I0 T I1
T
![Page 121: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/121.jpg)
Dynamical plane: n = 2
I1 I0
Both I0 and I1 are mapped one-to-oneover the entire bow-tie I0 T I1
T
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Dynamical plane: n = 2
So we have a preimage of the bow-tie inside each of I0 and I1
T
![Page 123: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/123.jpg)
Dynamical plane: n = 2
Then a second preimage, etc., etc.
T
![Page 124: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/124.jpg)
Dynamical plane: n = 2
The points whose orbits stay in I0 I1 form a Cantor set, and the preimages of T give the adjoined disks.
T
![Page 125: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/125.jpg)
Dynamical plane: n = 2
The points whose orbits stay in I0 I1 form a Cantor set, and the preimages of T give the adjoined disks.
QuickTime™ and a decompressor
are needed to see this picture.
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Cantor Necklaces in the Parameter Plane
€
Fλ (z) = z2 +λz2
cλ = λ1/4 vλ = 2 λ1/2 Fλ(vλ) = 1/4 + 4λ
D = {λ| |λ| < 1, Re(λ) < 0}
D
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For each λ D, have a Cantor set of points inside I1
I1
I0T. . . .. ...:. ..
![Page 128: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/128.jpg)
I1
I0T
For each λ D, have a Cantor set of points inside I1
. . . .. ...:. ..
Let zs(λ) be the point in the Cantor set with itinerary s
zs(λ)
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I1
I0T
For each λ D, have a Cantor set of points inside I1
. . . .. ...:. ..
Let zs(λ) be the point in the Cantor set with itinerary s
zs(λ)
zs(λ) depends analytically on λand continuously on s
![Page 130: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/130.jpg)
I1
I0T
For each λ D, have a Cantor set of points inside I1
. . . .. ...:. ..
Let zs(λ) be the point in the Cantor set with itinerary s
zs(λ)
and zs(λ) lies in the half-diskH given by |z| < 2, Re(z) < 0
H
zs(λ) depends analytically on λand continuously on s
![Page 131: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/131.jpg)
So have an analytic map λ zs(λ) that takes D into H
zs(λ)
DH
![Page 132: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/132.jpg)
So have an analytic map λ zs(λ) that takes D into H
zs(λ)
DH
Have another map λ G(λ) = Fλ(vλ) = 1/4 + 4λ which maps D over a larger half disk containing H
G(λ)
![Page 133: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/133.jpg)
So have an analytic map λ zs(λ) that takes D into H
zs(λ)
DH
Have another map λ G(λ) = Fλ(vλ) = 1/4 + 4λ which maps D over a larger half disk containing H
G-1
But G is invertible.
![Page 134: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/134.jpg)
So have an analytic map λ zs(λ) that takes D into H
zs(λ)
DH
Have another map λ G(λ) = Fλ(vλ) = 1/4 + 4λ which maps D over a larger half disk containing H
G-1
But G is invertible. So G-1(zs(λ)) maps D strictly inside itself.
![Page 135: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/135.jpg)
zs(λ)
DH
G-1
By the Schwarz Lemma, for each itinerary s there is a unique fixed point λs for the map G-1(zs(λ)).
λs
![Page 136: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/136.jpg)
zs(λ)
DH
This is a parameter for which G(λs) = zs(λs),i.e., the second iterate of the critical points lands
on a point in the Cantor set portion of the Cantor necklace.
G-1
By the Schwarz Lemma, for each itinerary s there is a unique fixed point λs for the map G-1(zs(λ)).
λs
zs(λs)
![Page 137: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/137.jpg)
zs(λ)
DH
G-1
So the points λs for each s give aCantor set of points in the parameter plane.
λs
zs(λs)
![Page 138: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/138.jpg)
zs(λ)
DH
G-1
So the points λs for each s give aCantor set of points in the parameter plane.
λs
Similar arguments involving Böttcher coordinates onand itineraries of preimages of the trap door
then append the Sierpinski holes to the necklace.
zs(λs)
![Page 139: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/139.jpg)
QuickTime™ and a decompressor
are needed to see this picture.QuickTime™ and a
decompressorare needed to see this picture.
This necklace lies along the negative real axis.
parameter plane n = 2
![Page 140: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/140.jpg)
parameter plane n = 2
There are lots of other Cantornecklaces in the parameter planes.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 141: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/141.jpg)
parameter plane n = 2
There are lots of other Cantornecklaces in the parameter planes.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 142: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/142.jpg)
parameter plane n = 2
There are lots of other Cantornecklaces in the parameter planes.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 143: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/143.jpg)
parameter plane n = 2
There are lots of other Cantornecklaces in the parameter planes.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
![Page 144: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/144.jpg)
When n > 2, get more complicated Cantor webscase n = 3:
![Page 145: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/145.jpg)
When n > 2, get more complicated Cantor webscase n = 3:
![Page 146: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/146.jpg)
When n > 2, get more complicated Cantor webscase n = 3:
![Page 147: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/147.jpg)
When n > 2, get more complicated Cantor webscase n = 3:
![Page 148: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/148.jpg)
When n > 2, get more complicated Cantor webscase n = 3:
Continue in this wayand then adjoin
Cantor sets
![Page 149: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/149.jpg)
QuickTime™ and a decompressor
are needed to see this picture.QuickTime™ and a
decompressorare needed to see this picture.
Parameter plane n = 3Dynamical plane
![Page 150: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/150.jpg)
QuickTime™ and a decompressor
are needed to see this picture.QuickTime™ and a
decompressorare needed to see this picture.
Parameter plane n = 3Dynamical plane
![Page 151: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/151.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
Parameter plane n = 3
QuickTime™ and a decompressor
are needed to see this picture.
Dynamical plane
![Page 152: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/152.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
Parameter plane n = 3
QuickTime™ and a decompressor
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Dynamical plane
![Page 153: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/153.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
Parameter plane n = 3Dynamical plane
QuickTime™ and a decompressor
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![Page 154: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/154.jpg)
Parameter plane n = 3
QuickTime™ and a decompressor
are needed to see this picture.
Cantor webs in the parameter plane
![Page 155: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/155.jpg)
Parameter plane n = 3
Cantor webs in the parameter plane
QuickTime™ and a decompressor
are needed to see this picture.
![Page 156: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/156.jpg)
Parameter plane n = 3
Cantor webs in the parameter plane
QuickTime™ and a decompressor
are needed to see this picture.QuickTime™ and a
decompressorare needed to see this picture.
Parameter plane
![Page 157: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/157.jpg)
Parameter plane n = 3
Cantor webs in the parameter plane
QuickTime™ and a decompressor
are needed to see this picture.QuickTime™ and a
decompressorare needed to see this picture.
Parameter plane
![Page 158: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/158.jpg)
Parameter plane n = 3
Cantor webs in the parameter plane
QuickTime™ and a decompressor
are needed to see this picture.QuickTime™ and a
decompressorare needed to see this picture.
Parameter plane
![Page 159: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/159.jpg)
Parameter plane n = 4
Different Cantor webs when n = 4
Dynamical plane
QuickTime™ and a decompressor
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QuickTime™ and a decompressor
are needed to see this picture.
![Page 160: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/160.jpg)
Part 2: Mandelpinski Necklaces
![Page 161: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/161.jpg)
Parameter plane for n = 3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centersof baby Mandelbrot sets and k centers of Sierpinski holes.
![Page 162: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/162.jpg)
Parameter plane for n = 3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centersof baby Mandelbrot sets and k centers of Sierpinski holes.
C1 passes through thecenters of 2 M-sets
and 2 S-holes
Easy check: C1 is the circle r = 2-2n/n-1
![Page 163: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/163.jpg)
Parameter plane for n = 3
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centersof baby Mandelbrot sets and k centers of Sierpinski holes.
![Page 164: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/164.jpg)
Parameter plane for n = 3
A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centersof baby Mandelbrot sets and k centers of Sierpinski holes.
C2 passes through thecenters of 4 M-sets
and 4 S-holesQuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
*
* only exception:2 centers of period 2 bulbs, not M-sets
![Page 165: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/165.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centersof baby Mandelbrot sets and k centers of Sierpinski holes.
Parameter plane for n = 3
C3 passes through thecenters of 10 M-sets
and 10 S-holes
![Page 166: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/166.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centersof baby Mandelbrot sets and k centers of Sierpinski holes.
Parameter plane for n = 3
C4 passes through thecenters of 28 M-sets
and 28 S-holes
![Page 167: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/167.jpg)
A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centersof baby Mandelbrot sets and k centers of Sierpinski holes.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Parameter plane for n = 3
C5 passes through thecenters of 82 M-sets
and 82 S-holes
![Page 168: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/168.jpg)
Theorem: There exist closed curves Cj, surrounding the McMullen domain. Each Cj passes
alternately through (n-2)nj-1 +1 centers of baby Mandelbrot sets and centers of Sierpinski holes.€
j = 1,...,∞
![Page 169: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/169.jpg)
C14 passes through thecenters of 4,782,969 M-sets and S-holesQuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
Parameter plane for n = 3
Theorem: There exist closed curves Cj, surrounding the McMullen domain. Each Cj passes
alternately through (n-2)nj-1 +1 centers of baby Mandelbrot sets and centers of Sierpinski holes.€
j = 1,...,∞
![Page 170: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/170.jpg)
Parameter plane for n = 4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
C1: 3 holes and M-sets
Theorem: There exist closed curves Cj, surrounding the McMullen domain. Each Cj passes
alternately through (n-2)nj-1 +1 centers of baby Mandelbrot sets and centers of Sierpinski holes.€
j = 1,...,∞
![Page 171: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/171.jpg)
Parameter plane for n = 4
C2: 9 holes and M-setsC3: 33 holes and M-setsQuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
Theorem: There exist closed curves Cj, surrounding the McMullen domain. Each Cj passes
alternately through (n-2)nj-1 +1 centers of baby Mandelbrot sets and centers of Sierpinski holes.€
j = 1,...,∞
*
![Page 172: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/172.jpg)
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Easy computations:
€
Fλ (z ) = z 3 +λz 3
€
λ =.08i
Critical points: λ1/2n
Prepoles: (-λ)1/2n
![Page 173: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/173.jpg)
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Easy computations:
€
Fλ (z ) = z 3 +λz 3
€
λ =.08i
All of the criticalpoints and prepoleslie on the “criticalcircle” : |z| = | |
€
λ 1/2n
€
γ0
€
γ0
![Page 174: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/174.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Fλ (z ) = z 3 +λz 3
€
λ =.08i
All of the criticalpoints and prepoleslie on the “criticalcircle” : |z| = | |
€
λ 1/2n
€
γ0
€
γ0
which is mapped 2n-to-1onto the “critical value line”
connecting
€
±vλ €
vλ
€
−vλ
Easy computations:
![Page 175: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/175.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Fλ (z ) = z 3 +λz 3
€
λ =.08i
€
γ0
€
vλ
Any other circle around 0is mapped n-to-1 to an ellipse
whose foci are
€
±2 λ
Easy computations:
€
−vλ
![Page 176: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/176.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Fλ (z ) = z 3 +λz 3
€
λ =.08i
€
γ0
€
vλ
€
±2 λ
Easy computations:
€
−vλ
Any other circle around 0is mapped n-to-1 to an ellipse
whose foci are
![Page 177: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/177.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Fλ (z ) = z 3 +λz 3
€
λ =.08i
€
γ0
€
vλ
€
±2 λ
So the exterior of is mapped as an n-to-1 covering of the
exterior of the critical value line. €
γ0
Easy computations:
€
−vλ
Any other circle around 0is mapped n-to-1 to an ellipse
whose foci are
![Page 178: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/178.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Fλ (z ) = z 3 +λz 3
€
λ =.08i
€
γ0
€
vλ
€
±2 λ
So the exterior of is mapped as an n-to-1 covering of the
exterior of the critical value line. Same with the interior of . €
γ0
€
γ0
Easy computations:
€
−vλ
Any other circle around 0is mapped n-to-1 to an ellipse
whose foci are
![Page 179: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/179.jpg)
Now assume that lies inside the critical circle:
€
±vλ
€
γ0€
vλ
€
−vλ
Warning: this is not a real proof....
![Page 180: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/180.jpg)
€
γ0
Now assume that lies inside the critical circle:
€
±vλ
The exterior of is mapped n-to-1onto the exterior of the critical value ray, so there is a preimage mapped n-to-1 to , €
γ0
€
γ0
€
γ1
€
vλ
€
−vλ
![Page 181: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/181.jpg)
Now assume that lies inside the critical circle:
€
±vλ
€
γ0€
vλ
€
−vλ
€
γ1
The exterior of is mapped n-to-1onto the exterior of the critical value ray, so there is a preimage mapped n-to-1 to , €
γ0
€
γ0
€
γ1
![Page 182: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/182.jpg)
then is mapped n-to-1 to ,
€
γ1
Now assume that lies inside the critical circle:
€
±vλ
€
γ2
€
γ0€
vλ
€
−vλ
€
γ1
€
γ2
The exterior of is mapped n-to-1onto the exterior of the critical value ray, so there is a preimage mapped n-to-1 to , €
γ0
€
γ0
€
γ1
![Page 183: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/183.jpg)
and on and onout to
€
∂B
Now assume that lies inside the critical circle:
€
±vλ
then is mapped n-to-1 to ,
€
γ1
€
γ0€
vλ
€
−vλ
€
γ1
€
γ2
B
The exterior of is mapped n-to-1onto the exterior of the critical value ray, so there is a preimage mapped n-to-1 to , €
γ0
€
γ0
€
γ1
€
γ2
![Page 184: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/184.jpg)
€
γ0 contains 2n critical points and 2n prepoles, so
€
γ1 contains 2n2 pre-critical points and pre-prepoles
€
γ0€
vλ
€
−vλ
€
γ1
![Page 185: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/185.jpg)
€
γ0 contains 2n critical points and 2n prepoles, so
€
γ1 contains 2n2 pre-critical points and pre-prepoles
€
γ0€
vλ
€
−vλ
€
γ1
€
γ2
B
€
γk contains 2nk+1 points thatmap to the critical pointsand pre-prepoles under
€
Fλk
![Page 186: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/186.jpg)
€
γ0 contains 2n critical points and 2n prepoles, so
€
γ1 contains 2n2 pre-critical points and pre-prepoles
€
γk contains 2nk+1 points thatmap to the critical pointsand pre-prepoles under
€
Fλk
n = 3
QuickTime™ and a decompressor
are needed to see this picture.
![Page 187: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/187.jpg)
€
γ0 contains 2n critical points and 2n prepoles, so
€
γ1 contains 2n2 pre-critical points and pre-prepoles
€
γk contains 2nk+1 points thatmap to the critical pointsand pre-prepoles under
€
Fλk
n = 3
QuickTime™ and a decompressor
are needed to see this picture.
![Page 188: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/188.jpg)
€
γ0 contains 2n critical points and 2n prepoles, so
€
γ1 contains 2n2 pre-critical points and pre-prepoles
€
γk contains 2nk+1 points thatmap to the critical pointsand pre-prepoles under
€
Fλk
n = 3
QuickTime™ and a decompressor
are needed to see this picture.
![Page 189: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/189.jpg)
€
γ0€
vλ
€
−vλ
€
γk
As rotates by one turn, these 2nk+1 points on each rotate by 1/2nk+1 of a turn.
€
γk
€
λ
![Page 190: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/190.jpg)
€
γ0€
vλ
€
−vλ
€
γk
Since
the second iterate of the criticalpoints rotate by 1 - n/2 ofa turn
As rotates by one turn, these 2nk+1 points on each rotate by 1/2nk+1 of a turn.
€
γk
€
λ
€
Fλ (vλ ) = 2nλn /2 +λ
2nλn /2 ≈12nλ1−n /2
![Page 191: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/191.jpg)
€
γ0€
vλ
€
−vλ
€
γk
€
Fλ (vλ ) = 2nλn /2 +λ
2nλn /2 ≈12nλ1−n /2
Since
the second iterate of the criticalpoints rotate by 1 - n/2 ofa turn, so this point hitsexactly
€
(n /2 −1)(2nk+1) +1 = (n − 2)nk+1 +1
preimages of the critical pointsand prepoles on
€
γk
As rotates by one turn, these 2nk+1 points on each rotate by 1/2nk+1 of a turn.
€
γk
€
λ
![Page 192: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/192.jpg)
There is a natural parametrization of each
€
γk
€
γkλ (θ )
€
γ0€
vλ
€
−vλ
€
γk
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€
γkλ (θ )
The real proof involves the Schwarz Lemma (as before):
![Page 193: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/193.jpg)
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There is a natural parametrization of each
€
γk
€
γkλ (θ )
The real proof involves the Schwarz Lemma (as before):
€
γ0€
vλ
€
−vλ
€
γk
€
γkλ (θ )
Best to restrict to a “symmetry region” inside the circle C1, so that is well-defined.
€
γkλ (θ )
![Page 194: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/194.jpg)
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€
γ0€
vλ
€
−vλ
€
γk
€
γkλ (θ )
Best to restrict to a “symmetry region” inside the circle C1, so that is well-defined.
Then we have a second map from the parameter plane to thedynamical plane, namely which is invertible on the symmetry sector
€
G(λ ) = Fλ (vλ )
€
G−1
€
γkλ (θ )
![Page 195: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/195.jpg)
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€
γ0€
vλ
€
−vλ
€
γk
€
γkλ (θ )
Then we have a second map from the parameter plane to thedynamical plane, namely which is invertible on the symmetry sector
€
G(λ ) = Fλ (vλ )
€
G−1(γ kλ (θ ) )
a map from a “disk” to itself.
So consider the composition€
G−1
![Page 196: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/196.jpg)
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€
γ0€
vλ
€
−vλ
€
γk
€
γkλ (θ )
€
G−1(γ kλ (θ ) )
a map from a “disk” to itself.
So consider the composition
Schwarz implies that has a unique fixed point,i.e., a parameter for which the second iterate of the criticalpoint lands on the point , so this proves theexistence of lots of parameters for which the critical orbits are periodic and land on 0.
€
G−1(γ kλ (θ ) )
€
γkλ (θ )
€
G−1
![Page 197: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/197.jpg)
Remarks:
1. This proves the existence of centers of Sierpinski holes and Mandelbrot sets. Producingthe entire M-set involves polynomial-like maps;while the entire S-hole involves qc-surgery.
![Page 198: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/198.jpg)
Remarks:
1. This proves the existence of centers of Sierpinski holes and Mandelbrot sets. Producingthe entire M-set involves polynomial-like maps;while the entire S-hole involves qc-surgery.
2. It is known that each S-hole in the Mandelpinskinecklace is also surrounded by infinitely many sub-necklaces, which in turn are surrounded bysub-sub-necklaces, etc.
![Page 199: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/199.jpg)
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n = 3
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3. CanManPinski Trees
A tree of Cantor necklaces with Mandelbrot sets interspersed between each branch
parameter plane parameter plane
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n = 2
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Dynamical plane: n = 2
I1
I0
Recall that we have a Cantor necklace in the dynamical plane lying in I0 T I1
![Page 202: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/202.jpg)
Dynamical plane: n = 2
I1
I0
The regions I2 and I3 are mapped one-to-one overI0 T I1, so there are Cantor necklaces in I2 and I3
I2
I3
![Page 203: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/203.jpg)
Dynamical plane: n = 2
I1
I0
I2
I3
The regions I2 and I3 are mapped one-to-one overI0 T I1, so there are Cantor necklaces in I2 and I3
![Page 204: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/204.jpg)
This necklace is mapped one-to-one ontothe original necklace.
![Page 205: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/205.jpg)
This necklace is mapped one-to-one ontothe original necklace.
And so is the bottom necklace
![Page 206: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/206.jpg)
S2
S0
S1
S3
Now consider the regions Sj.
![Page 207: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/207.jpg)
S2
S0
S1
S3
Now consider the regions Sj.
S0 is mapped two-to-one onto S0 S1
![Page 208: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/208.jpg)
S2
S0
S1
S3
Now consider the regions Sj.
S0 is mapped two-to-one onto S0 S1
Similarly, S1 S2 S3,S2 S0 S1
and S3 S2 S3
![Page 209: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/209.jpg)
S2
S0
S1
S3
Assuming λ lies in the upper half plane, the critical values ±vλ lie in S0 and S2 (easy check)
vλ
-vλ
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S2
S0
S1
S3
Assuming λ lies in the upper half plane, the critical values ±vλ lie in S0 and S2 (easy check)So there is a region in S3 mapped one-to-one ontoS3.
vλ
-vλ
![Page 211: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/211.jpg)
S2
S0
S1
S3
So there is a preimage of this Cantor necklace in S3
vλ
-vλ
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S2
S0
S1
S3
So there is a preimage of this Cantor necklace in S3,
vλ
-vλ
![Page 213: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/213.jpg)
S2
S0
S1
S3
So there is a preimage of this Cantor necklace in S3,and then another preimage,
vλ
-vλ
![Page 214: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/214.jpg)
S2
S0
S1
S3
So there is a preimage of this Cantor necklace in S3,and then another preimage, and so on, yielding infinitelymany necklaces eventually mapping to the original necklace. Looking like branchesof a tree....
vλ
-vλ
![Page 215: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/215.jpg)
S2
S0
S1
S3
By symmetry, we have similar branches in S1
vλ
-vλ
![Page 216: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/216.jpg)
S2
S0
S1
S3
By symmetry, we have similar branches in S1, S0,
![Page 217: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/217.jpg)
S2
S0
S1
S3
By symmetry, we have similar branches in S1, S0, and S2
![Page 218: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/218.jpg)
This produces trees of Cantor necklaces in the dynamical plane
![Page 219: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/219.jpg)
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This produces trees of Cantor necklaces in the dynamical plane
![Page 220: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/220.jpg)
Assuming λ is in the upper half plane, we can again use G(λ) = 1/4 + 4 λ and an appropriate coding of points in the necklace, and then the Schwarz Lemma produces a similar
tree in the upper half of the parameter plane.
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Symmetry under complex conjugation yields a similar tree in the lower half-plane.
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Then polynomial-like map theory produces a Mandelbrot set in each region in between the branches.
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Open problems:
Can we use these trees to give a complete map of the Sierpinski-hole regions and buried parameters in the parameter planes?
Same question for the baby Mandelbrot sets.
Using “Yoccoz puzzles,” do these trees allow us tosee that the boundary of the parameter plane locusis a simple closed curve?
![Page 224: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/224.jpg)
Open problems:
Can we use these trees to give a complete map of the Sierpinski-hole regions and buried parameters in the parameter planes?
Same question for the baby Mandelbrot sets.
Using “Yoccoz puzzles,” do these trees allow us tosee that the boundary of the parameter plane locusis a simple closed curve?
Would it be better to call these thingsCantormandelbrotsierpinski trees?
![Page 225: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/225.jpg)
Open problems:
Can we use these trees to give a complete map of the Sierpinski-hole regions and buried parameters in the parameter planes?
Same question for the baby Mandelbrot sets.
Using “Yoccoz puzzles,” do these trees allow us tosee that the boundary of the parameter plane locusis a simple closed curve?
Would it be better to call these thingsCantormandelbrotsierpinski trees?
Who the hell is this?
![Page 226: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/226.jpg)
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Quick
Tim
e™ a
nd a
TIFF
(LZW
) dec
ompr
esso
rar
e ne
eded
to se
e th
is pi
ctur
e.Parameter plane (rotated)
when n = 2
![Page 227: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/227.jpg)
Other topics:
Main cardioid of a buried baby M-set
Perturbed rabbit
Convergence to the unit disk
Major application
![Page 228: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/228.jpg)
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
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n = 4
![Page 229: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/229.jpg)
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n = 4
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 230: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/230.jpg)
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n = 4
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If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 231: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/231.jpg)
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n = 4
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A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 232: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/232.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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n = 4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 233: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/233.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 234: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/234.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 4
QuickTime™ and aTIFF (LZW) decompressor
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A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 235: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/235.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 236: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/236.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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n = 4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 237: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/237.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 238: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/238.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 239: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/239.jpg)
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n = 4
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A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 240: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/240.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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n = 4
QuickTime™ and aTIFF (LZW) decompressor
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A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 241: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/241.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
n = 4
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 242: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/242.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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n = 4
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A Sierpinski curve, but very different dynamically from the earlier ones.
![Page 243: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/243.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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n = 4
QuickTime™ and aTIFF (LZW) decompressor
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A Sierpinski curve, but very different dynamically from the earlier ones.
If λ lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.
![Page 244: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/244.jpg)
Consider the family of maps
where c is the center of a hyperbolic component of the Mandelbrot set.
€
Fλ (z) = zn + c+ λzn
![Page 245: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/245.jpg)
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λ =0
€
c = −1
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Consider the family of maps
where c is the center of a hyperbolic component of the Mandelbrot set.
€
Fλ (z) = zn + c+ λzn
![Page 246: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/246.jpg)
€
c = −.12 +.75i
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λ =0
Consider the family of maps
where c is the center of a hyperbolic component of the Mandelbrot set.
€
Fλ (z) = zn + c+ λzn
![Page 247: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/247.jpg)
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λ ≠0, the Julia set again expodes.When
![Page 248: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/248.jpg)
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λ ≠0, the Julia set again expodes.When
![Page 249: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/249.jpg)
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λ ≠0, the Julia set again expodes.When
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![Page 250: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/250.jpg)
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λ ≠0, the Julia set again expodes.When
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![Page 251: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/251.jpg)
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λ ≠0, the Julia set again expodes.When
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A doubly-invertedDouady rabbit.
![Page 252: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/252.jpg)
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If you chop off the “ears” of each internal rabbit in each component of the original Julia set, then what’s left is another Sierpinski curve (provided that both of the critical orbits eventually escape).
![Page 253: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/253.jpg)
![Page 254: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/254.jpg)
The case n = 2 is very different from (and much more difficult than) the case n > 2.
n = 3 n = 2
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![Page 255: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/255.jpg)
One difference: there is a McMullen domain whenn > 2, but no McMullen domain when n = 2
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n = 3 n = 2
![Page 256: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/256.jpg)
One difference: there is a McMullen domain whenn > 2, but no McMullen domain when n = 2
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n = 3 n = 2
![Page 257: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/257.jpg)
There is lots of structure when n > 2, but what is going on when n = 2?
n = 3 n = 2
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![Page 258: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/258.jpg)
There is lots of structure when n > 2, but what is going on when n = 2?
n = 3 n = 2
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![Page 259: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/259.jpg)
There is lots of structure when n > 2, but what is going on when n = 2?
n = 3 n = 2
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![Page 260: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/260.jpg)
Also, not much is happening for the Julia sets near 0 when n > 2
n = 3
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λ =.01
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![Page 261: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/261.jpg)
The Julia set is always aCantor set of circles.
n = 3
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λ =.0001
![Page 262: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/262.jpg)
n = 3
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The Julia set is always aCantor set of circles.
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λ =.000001
![Page 263: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/263.jpg)
The Julia set is always aCantor set of circles.
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λ =.000001
There is always a round annulus of some fixed width in the Fatou set,
so the Julia set does not convergeto the unit disk.
![Page 264: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/264.jpg)
n = 2
But when n = 2, lots of things happen near the origin;in fact, the Julia sets converge to the unit disk as
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λ → 0
disk-converge
![Page 265: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/265.jpg)
Here’s the parameter plane when n = 2:
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![Page 266: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/266.jpg)
Quick
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Rotate it by 90 degrees:
and this object appears everywhere.....
![Page 267: Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b267f8b9ab0599970f1/html5/thumbnails/267.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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