dynamics of the family of complex maps paul blanchard toni garijo matt holzer u. hoomiforgot dan...
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![Page 1: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/1.jpg)
Dynamics of the family of complex maps
Paul BlanchardToni GarijoMatt HolzerU. HoomiforgotDan LookSebastian Marotta
with:€
Fλ (z) = zn +λ
zn, n ≥ 2
(why the case n = 2 is )
Mark MorabitoMonica Moreno RochaKevin PilgrimElizabeth RussellYakov ShapiroDavid Uminsky
CR AZ Y
![Page 2: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/2.jpg)
€
Fλ (z) = z n +λ
z n
The case n > 2 is great because:
![Page 3: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/3.jpg)
€
Fλ (z) = z n +λ
z n
The case n > 2 is great because:
There exists a McMullen domain around = 0 ....
QuickTime™ and aTIFF (LZW) decompressor
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QuickTime™ and aTIFF (LZW) decompressor
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Parameter plane for n=3
€
λ
![Page 4: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/4.jpg)
€
Fλ (z) = z n +λ
z n
The case n > 2 is great because:
... surrounded by infinitely many “Mandelpinski” necklaces...
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QuickTime™ and aTIFF (LZW) decompressor
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There exists a McMullen domain around = 0 ....
€
λ
![Page 5: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/5.jpg)
€
Fλ (z) = z n +λ
z n
The case n > 2 is great because:
... surrounded by infinitely many “Mandelpinski” necklaces...
... and the Julia sets behave nicely as
€
λ → 0
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€
λ =.01
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€
λ =.0001
€
λ =.000001
There exists a McMullen domain around = 0 ....
€
λ
![Page 6: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/6.jpg)
€
Fλ (z) = z n +λ
z n
There is no McMullen domain....
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
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The case n = 2 is crazy because:
![Page 7: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/7.jpg)
€
Fλ (z) = z n +λ
z n
There is no McMullen domain....
... and no “Mandelpinski” necklaces...
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
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The case n = 2 is crazy because:
![Page 8: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/8.jpg)
€
Fλ (z) = z n +λ
z n
There is no McMullen domain....
... and the Julia sets “go crazy” as
€
λ → 0
... and no “Mandelpinski” necklaces...
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QuickTime™ and aTIFF (LZW) decompressor
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QuickTime™ and aTIFF (LZW) decompressor
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€
λ =−.000001
€
λ =−.0001
€
λ =−.01
The case n = 2 is crazy because:
![Page 9: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/9.jpg)
Some definitions:
Julia set of
J = boundary of {orbits that escape to }
€
∞
= closure {repelling periodic orbits}
= {chaotic set}
Fatou set
= complement of J
= predictable set
€
Fλ (z) = zn +λ
zn
![Page 10: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/10.jpg)
Computation of J:
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Color points that escape toinfinity shades of red orange yellow green blue violet Black points do not escape.J = boundary of the black region.
€
λ =.08i
€
Fλ (z ) = z 3 +λ
z 3
![Page 11: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/11.jpg)
Easy computations:
€
λ =.08i
is superattracting, so have immediate basin Bmapped n-to-1 to itself.
€
∞€
Fλ (z ) = z 3 +λ
z 3
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B
![Page 12: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/12.jpg)
Easy computations:
is superattracting, so have immediate basin Bmapped n-to-1 to itself.
€
∞€
Fλ (z ) = z 3 +λ
z 3
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B
T
€
λ =.08i
0 is a pole, so havetrap door T mapped
n-to-1 to B.
![Page 13: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/13.jpg)
Easy computations:
is superattracting, so have immediate basin Bmapped n-to-1 to itself.
€
∞€
Fλ (z ) = z 3 +λ
z 3
0 is a pole, so havetrap door T mapped
n-to-1 to B.
The Julia set has 2n-fold symmetry.
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B
T
€
λ =.08i
![Page 14: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/14.jpg)
Easy computations:
€
Fλ (z ) = z 3 +λ
z 3
2n free critical points
€
cλ = λ1/2n
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€
λ =.08i
![Page 15: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/15.jpg)
Easy computations:
2n free critical points
€
cλ = λ1/2n
€
Fλ (z ) = z 3 +λ
z 3
QuickTime™ and aTIFF (LZW) decompressor
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€
λ =.08i
![Page 16: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/16.jpg)
Easy computations:
2n free critical points
€
cλ = λ1/2n
Only 2 critical values
€
vλ = ±2 λ
€
Fλ (z ) = z 3 +λ
z 3
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€
λ =.08i
![Page 17: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/17.jpg)
Easy computations:
2n free critical points
€
cλ = λ1/2n
Only 2 critical values
€
vλ = ±2 λ
€
Fλ (z ) = z 3 +λ
z 3
€
λ =.08i
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![Page 18: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/18.jpg)
Easy computations:
2n free critical points
€
cλ = λ1/2n
Only 2 critical values
€
vλ = ±2 λ
€
Fλ (z ) = z 3 +λ
z 3
€
λ =.08i
QuickTime™ and aTIFF (LZW) decompressor
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![Page 19: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/19.jpg)
Easy computations:
2n free critical points
€
cλ = λ1/2n
Only 2 critical values
€
vλ = ±2 λ
But really only 1 freecritical orbit
€
Fλ (z ) = z 3 +λ
z 3
€
λ =.08i
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![Page 20: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/20.jpg)
Easy computations:
2n free critical points
€
pλ = (−λ )1/2n
Only 2 critical values
€
vλ = ±2 λ
But really only 1 freecritical orbit
€
Fλ (z ) = z 3 +λ
z 3
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€
λ =.08iAnd 2n prepoles
€
cλ = λ1/2n
![Page 21: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/21.jpg)
The Escape Trichotomy
There are three possible ways that thecritical orbits can escape to infinity,
and each yields a different typeof Julia set.
(with D. Look & D Uminsky)
![Page 22: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/22.jpg)
The Escape Trichotomy
€
vλ∈
€
J ( Fλ
)
€
⇒B is a Cantor set
(with D. Look & D Uminsky)
![Page 23: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/23.jpg)
The Escape Trichotomy
€
vλ∈
€
J ( Fλ
)
€
⇒B
€
⇒
is a Cantor set
T
€
vλ∈ is a Cantor set of
simple closed curves
€
J ( Fλ
)
(McMullen)
(with D. Look & D Uminsky)
(n > 2)
![Page 24: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/24.jpg)
The Escape Trichotomy
€
vλ∈
€
J ( Fλ
)
€
⇒B
€
⇒€
⇒
is a Cantor set
T
€
vλ∈ is a Cantor set of
simple closed curves
€
J ( Fλ
)
€
Fλ
k(v
λ) ∈ T
€
J ( Fλ
) is a Sierpinski curve
(McMullen)
(with D. Look & D Uminsky)
(n > 2)
In all other cases is a connected set, and if
€
J ( Fλ
)
![Page 25: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/25.jpg)
€
vλ∈
€
J ( Fλ
)
€
⇒BCase 1:
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parameter planewhen n = 3
is a Cantor set
![Page 26: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/26.jpg)
€
vλ∈
€
J ( Fλ
)
€
⇒BCase 1:
QuickTime™ and aTIFF (LZW) decompressor
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parameter planewhen n = 3
€
λ
is a Cantor set
![Page 27: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/27.jpg)
€
vλ∈
€
J ( Fλ
)
€
⇒BCase 1:
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parameter planewhen n = 3
QuickTime™ and aTIFF (LZW) decompressor
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J is a Cantor set
€
λ
is a Cantor set
![Page 28: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/28.jpg)
€
vλ∈
€
J ( Fλ
)
€
⇒BCase 1:
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parameter planewhen n = 3
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J is a Cantor set
€
λ
€
cλ
€
vλ
is a Cantor set
![Page 29: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/29.jpg)
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parameter planewhen n = 3
Case 2: T
€
vλ∈
€
⇒
€
J ( Fλ
) is a Cantor set ofsimple closed curves
![Page 30: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/30.jpg)
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parameter planewhen n = 3
€
λ
Case 2: T
€
vλ∈
€
⇒
€
J ( Fλ
) is a Cantor set ofsimple closed curves
![Page 31: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/31.jpg)
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The central disk isthe McMullen domain
€
λ
Case 2: T
€
vλ∈ is a Cantor set of
simple closed curves
€
⇒
€
J ( Fλ
)
![Page 32: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/32.jpg)
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parameter planewhen n = 3
€
λ
Case 2: T
€
vλ∈ is a Cantor set of
simple closed curves
J is a Cantor set of simple closed curves
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€
⇒
€
J ( Fλ
)
T
B
![Page 33: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/33.jpg)
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parameter planewhen n = 3
€
λ
Case 2: T
€
vλ∈ is a Cantor set of
simple closed curves
J is a Cantor set of simple closed curves
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€
⇒
€
J ( Fλ
)
€
cλ
€
vλ
![Page 34: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/34.jpg)
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parameter planewhen n = 3
Case 3:
€
⇒
€
Fλk(v λ ) ∈ T
€
J ( Fλ
) is a Sierpinski curve
![Page 35: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/35.jpg)
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parameter planewhen n = 3
Case 3:
€
⇒
€
Fλk(v λ ) ∈ T
€
J ( Fλ
)
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A Sierpinski curve is a planarset homeomorphic to theSierpinski carpet fractal
is a Sierpinski curve
![Page 36: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/36.jpg)
Sierpinski curves are important for two reasons:
1. There is a “topological characterization” of the carpet
2. A Sierpinski curve is a “universal plane continuum”
![Page 37: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/37.jpg)
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The Sierpinski Carpet
Topological Characterization
Any planar set that is:1. compact2. connected3. locally connected4. nowhere dense5. any two complementary domains are bounded by simple closed curves that are pairwise disjoint is a Sierpinski curve.
![Page 38: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/38.jpg)
Universal Plane Continuum
Any planar, one-dimensional, compact, connected set can be homeomorphically embedded in a Sierpinski curve.
For example....
![Page 39: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/39.jpg)
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This set
![Page 40: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/40.jpg)
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This set
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can be embedded inside
![Page 41: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/41.jpg)
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parameter planewhen n = 3
Case 3:
€
⇒
€
Fλk(v λ ) ∈ T
€
J ( Fλ
) is a Sierpinski curve
![Page 42: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/42.jpg)
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€
λ
Case 3:
€
⇒
€
Fλk(v λ ) ∈ T
€
J ( Fλ
) is a Sierpinski curve
A Sierpinski “hole”
![Page 43: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/43.jpg)
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€
λ
Case 3:
€
⇒
€
Fλk(v λ ) ∈ T
€
J ( Fλ
)
A Sierpinski curve
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is a Sierpinski curve
A Sierpinski “hole”
![Page 44: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/44.jpg)
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€
λ
Case 3:
€
⇒
€
Fλk(v λ ) ∈ T
€
J ( Fλ
)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
cλ
€
vλ
€
Fλ (vλ )
A Sierpinski curve
is a Sierpinski curve
A Sierpinski “hole”Escape time 3
![Page 45: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/45.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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€
λ
Case 3:
€
⇒
€
Fλk(v λ ) ∈ T
€
J ( Fλ
)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Sierpinski curve
is a Sierpinski curve
Another Sierpinski “hole”
![Page 46: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/46.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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€
λ
Case 3:
€
⇒
€
Fλk(v λ ) ∈ T
€
J ( Fλ
)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Sierpinski curve
is a Sierpinski curve
Another Sierpinski “hole”
€
cλ
€
vλ
€
Fλ2(vλ )
Escape time 4
![Page 47: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/47.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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Case 3:
€
⇒
€
Fλk(v λ ) ∈ T
€
J ( Fλ
)
€
λ
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Sierpinski curve
is a Sierpinski curve
Another Sierpinski “hole”Escape time 7
![Page 48: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/48.jpg)
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Case 3:
€
⇒
€
Fλk(v λ ) ∈ T
€
J ( Fλ
)
€
λ
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
A Sierpinski curve
is a Sierpinski curve
Another Sierpinski “hole”Escape time 5
![Page 49: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/49.jpg)
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So to show that is homeomorphic to
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![Page 50: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/50.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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Need to show:
compactconnectednowhere denselocally connectedbounded by disjoint s.c.c.’s
![Page 51: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/51.jpg)
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Need to show:
compactconnectednowhere denselocally connectedbounded by disjoint s.c.c.’s
Fatou set is the union of the preimages of B; all disjoint, open disks.
![Page 52: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/52.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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Need to show:
compactconnectednowhere denselocally connectedbounded by disjoint s.c.c.’s
Fatou set is the union of the preimages of B; all disjoint, open disks.
![Page 53: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/53.jpg)
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Need to show:
compactconnectednowhere denselocally connectedbounded by disjoint s.c.c.’s
If J contains an open set, then J = C.
![Page 54: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/54.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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Need to show:
compactconnectednowhere denselocally connectedbounded by disjoint s.c.c.’s
If J contains an open set, then J = C.
![Page 55: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/55.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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Need to show:
compactconnectednowhere denselocally connectedbounded by disjoint s.c.c.’s
No recurrent critical orbits and no parabolic points.
![Page 56: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/56.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Need to show:
compactconnectednowhere denselocally connectedbounded by disjoint s.c.c.’s
No recurrent critical orbits and no parabolic points.
![Page 57: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/57.jpg)
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Need to show:
compactconnectednowhere denselocally connectedbounded by disjoint s.c.c.’s
J locally connected, so theboundaries are locally connected. Need to show they are s.c.c.’s. Can only meet at (preimages of) critical points, hence disjoint.
![Page 58: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/58.jpg)
QuickTime™ and aTIFF (LZW) decompressor
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Need to show:
compactconnectednowhere denselocally connectedbounded by disjoint s.c.c.’s
So J is a Sierpinski curve.
![Page 59: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/59.jpg)
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Remark: All Julia sets drawn from Sierpinski holes are homeomorphic, but only those in symmetrically locatedSierpinski holes have the same dynamics.
The maps on all these Julia sets are dynamically different.
![Page 60: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/60.jpg)
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Remark: All Julia sets drawn from Sierpinski holes are homeomorphic, but only those in symmetrically locatedSierpinski holes have the same dynamics.
In fact, there are exactly (n-1)(2n)k-3 Sierpinski holes withescape time k, and (2n)k-3 different conjugacy classes (n odd).
The maps on all these Julia sets are dynamically different.
(with K.Pilgrim)
![Page 61: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/61.jpg)
1. The McMullen domain
2. Mandelpinski necklaces
3. Julia sets near 0
Topics
![Page 62: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/62.jpg)
Part 1: The McMullen Domain
![Page 63: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/63.jpg)
Why is there no McMullen domain when n = 2?
What is the preimage of T?
First suppose
€
vλ∈ T:
![Page 64: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/64.jpg)
Why is there no McMullen domain when n = 2?
First suppose
What is the preimage of T?
€
vλ∈ T:
Can the preimage be 2n disjoint disks, each of which contains a critical point?
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Why is there no McMullen domain when n = 2?
First suppose
What is the preimage of T?
€
vλ∈ T:
Can the preimage be 2n disjoint disks, each of which contains a critical point?
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No --- there would then be 4n preimages of any point in T, but the map has degree 2n.
![Page 66: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/66.jpg)
Why is there no McMullen domain when n = 2?
So some of the preimages of Tmust overlap, and by 2n-foldsymmetry, all must intersect.
![Page 67: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/67.jpg)
Why is there no McMullen domain when n = 2?
So some of the preimages of Tmust overlap, and by 2n-foldsymmetry, all must intersect.
By Riemann-Hurwitz, the preimage of T must then
be an annulus.
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€
cλ
€
vλ
![Page 68: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/68.jpg)
Why is there no McMullen domain when n = 2?
So here is the picture:
T
B
A
A is the annulus separating B and T
![Page 69: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/69.jpg)
Why is there no McMullen domain when n = 2?
So here is the picture:B
TA
A is the annulus separating B and T
XF maps X 2n-to-1
onto T
![Page 70: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/70.jpg)
Why is there no McMullen domain when n = 2?
So here is the picture:B
TA
A is the annulus separating B and T
X
A
A
0
1
F maps both A0
1and A as an n-to-1
covering onto A
F maps X 2n-to-1onto T
![Page 71: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/71.jpg)
Why is there no McMullen domain when n = 2?
B
TA
X
A
A
0
1
So mod(A0) = 1/n mod(A)
And mod(A1) = 1/n mod(A)
![Page 72: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/72.jpg)
Why is there no McMullen domain when n = 2?
B
TA
X
A
A
0
1
So mod(A0) = 1/n mod(A)
And mod(A1) = 1/n mod(A)
When n = 2,mod(A0) + mod(A1) =
mod(A)
![Page 73: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/73.jpg)
Why is there no McMullen domain when n = 2?
B
TA
X
A
A
0
1
So mod(A0) = 1/n mod(A)
And mod(A1) = 1/n mod(A)
When n = 2,mod(A0) + mod(A1) =
mod(A)
So there is no room for X, i.e., does
not lie in T
€
vλ
![Page 74: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/74.jpg)
€
vλ = ±2 λ
€
Fλ (vλ ) = 2nλn /2 +λ
2nλn /2 = 2nλn /2 +1
2nλ (n /2−1)
Why is there no McMullen domain when n = 2?
Here is another reason:
![Page 75: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/75.jpg)
€
Fλ (vλ ) = 2nλn /2 +λ
2nλn /2 = 2nλn /2 +1
2nλ (n /2−1)
Why is there no McMullen domain when n = 2?
Here is another reason:
€
n > 2⇒ Fλ (vλ ) → ∞ as λ → 0
€
vλso lies in T when n > 2
€
vλ = ±2 λ
![Page 76: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/76.jpg)
Why is there no McMullen domain when n = 2?
Here is another reason:
€
n > 2⇒ Fλ (vλ ) → ∞ as λ → 0
€
n = 2⇒ Fλ (vλ ) = 4λ +1
4€
vλso lies in T when n > 2
€
vλ = ±2 λ
€
Fλ (vλ ) = 2nλn /2 +λ
2nλn /2 = 2nλn /2 +1
2nλ (n /2−1)
![Page 77: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/77.jpg)
€
Fλ (vλ ) = 2nλn /2 +λ
2nλn /2 = 2nλn /2 +1
2nλ (n /2−1)
Why is there no McMullen domain when n = 2?
Here is another reason:
€
n > 2⇒ Fλ (vλ ) → ∞ as λ → 0
€
so Fλ (vλ ) →1/ 4 as λ → 0(???)€
vλso lies in T when n > 2
€
vλ = ±2 λ
€
n = 2⇒ Fλ (vλ ) = 4λ +1
4
![Page 78: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/78.jpg)
Part 2: Mandelpinski Necklaces
![Page 79: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/79.jpg)
Parameter plane for n = 3
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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 80: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/80.jpg)
Parameter plane for n = 3
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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
![Page 81: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/81.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 82: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/82.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
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*
* only exception:2 centers of period 2 bulbs, not M-sets
![Page 83: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/83.jpg)
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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.
C3 passes through thecenters of 10 M-sets
and 10 S-holes
Parameter plane for n = 3
![Page 84: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/84.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.
C4 passes through thecenters of 28 M-sets
and 28 S-holes
Parameter plane for n = 3
![Page 85: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/85.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.
C5 passes through thecenters of 82 M-sets
and 82 S-holesQuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
Parameter plane for n = 3
![Page 86: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/86.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 87: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/87.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 88: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/88.jpg)
Parameter plane for n = 4
QuickTime™ and aTIFF (LZW) decompressor
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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 89: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/89.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 90: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/90.jpg)
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Easy computations:
€
Fλ (z ) = z 3 +λ
z 3
€
λ =.08i
All of the criticalpoints and prepoleslie on the “criticalcircle” : |z| = | |
€
λ 1/2n
€
γ0
€
γ0
![Page 91: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/91.jpg)
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are needed to see this picture.
€
Fλ (z ) = z 3 +λ
z 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 92: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/92.jpg)
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are needed to see this picture.
€
Fλ (z ) = z 3 +λ
z 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 93: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/93.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Fλ (z ) = z 3 +λ
z 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 94: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/94.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Fλ (z ) = z 3 +λ
z 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 95: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/95.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
Fλ (z ) = z 3 +λ
z 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 96: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/96.jpg)
Simplest case: C1
Assume that both sit on the critical circle.
€
±vλ
€
vλ
€
−vλ
€
γ0
€
pλ
(i.e., )
€
| vλ | = | cλ | = | pλ |
€
⇒ 2 | λ |1/2 = | λ |1/2n
![Page 97: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/97.jpg)
Assume that both sit on the critical circle.
€
±vλ
€
vλ
€
−vλ
€
γ0
€
pλ
Simplest case: C1
€
⇒ 2 | λ |1/2 = | λ |1/2n
€
⇒ 22n | λ |n = | λ |
€
⇒ | λ | = 2−
2n
n−1 ⎛ ⎝
⎞ ⎠
![Page 98: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/98.jpg)
Assume that both sit on the critical circle.
€
±vλ
This is the “dividing circle” in the parameter plane
Parameter plane n = 3
QuickTime™ and aTIFF (LZW) decompressor
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r = 2-3
Simplest case: C1
€
⇒ 2 | λ |1/2 = | λ |1/2n
€
⇒ 22n | λ |n = | λ |
€
⇒ | λ | = 2−
2n
n−1 ⎛ ⎝
⎞ ⎠
![Page 99: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/99.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Assume that both sit on the critical circle.
€
±vλ
€
⇒ 2 | λ |1/2 = | λ |1/2n
€
⇒ 22n | λ |n = | λ |
This is the “dividing circle” in the parameter plane
Parameter plane n = 4
€
⇒ | λ | = 2−
2n
n−1 ⎛ ⎝
⎞ ⎠
r = 2-8/3
Simplest case: C1
![Page 100: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/100.jpg)
Assume that lies on the dividing circle,so both sit on the critical circle.
€
±vλ
€
vλ
€
−vλ
€
γ0
€
λ
€
pλ
In this picture, is real and n = 3
€
λ
Simplest case: C1
![Page 101: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/101.jpg)
Assume that lies on the dividing circle,so both sit on the critical circle.
€
±vλ
€
vλ
€
−vλ
€
γ0
As rotates around the dividing circle, rotates a half-turn, while and rotate1/2n of a turn. So each meets exactlyn - 1 prepoles and critical points.€
λ
€
±vλ
€
pλ
€
vλ
€
λ
€
pλ€
cλ
In this picture, is real and n = 3
€
λ
Simplest case: C1
![Page 102: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/102.jpg)
Assume that lies on the dividing circle,so both sit on the critical circle.
€
±vλ
€
vλ
€
−vλ
€
γ0
As rotates around the dividing circle, rotates a half-turn, while and rotate1/2n of a turn. So each meets exactlyn - 1 prepoles and critical points.€
λ
€
±vλ
€
pλ
€
vλ
€
λ
€
pλ€
cλ
In this picture, is real and n = 3
€
λ
Simplest case: C1
So the dividing circle is C1
![Page 103: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/103.jpg)
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The dividing circle when n = 5
n-1 = 4 centers of Sierpinski holes;n-1 = 4 centers of baby Mandelbrot sets
![Page 104: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/104.jpg)
Now assume that lies inside the critical circle:
€
±vλ
€
γ0€
vλ
€
−vλ
![Page 105: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/105.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 106: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/106.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 107: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/107.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 108: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/108.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 109: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/109.jpg)
€
γ0 contains 2n critical points and 2n prepoles, so
€
γ1 contains 2n2 pre-critical points and pre-prepoles
€
γ0€
vλ
€
−vλ
€
γ1
![Page 110: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/110.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 111: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/111.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 112: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/112.jpg)
€
γ0€
vλ
€
−vλ
€
γk
€
Fλ (vλ ) ≈λ
2nλn /2 =1
2nλ1−n /2
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
![Page 113: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/113.jpg)
€
γ0€
vλ
€
−vλ
€
γk
€
Fλ (vλ ) ≈λ
2nλn /2 =1
2nλ1−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 114: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/114.jpg)
There is a natural parametrization of each
€
γk
€
γkλ (θ )
The real proof involves the Schwarz Lemma:
€
γ0€
vλ
€
−vλ
€
γk
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
γkλ (θ )
![Page 115: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/115.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
There is a natural parametrization of each
€
γk
€
γkλ (θ )
The real proof involves the Schwarz Lemma:
€
γ0€
vλ
€
−vλ
€
γk
€
γkλ (θ )
Best to restrict to a “symmetry region” inside the dividingcircle, so that is well-defined.
€
γkλ (θ )
![Page 116: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/116.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
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
Best to restrict to a “symmetry region” inside the dividingcircle, so that is well-defined.
€
γkλ (θ )
![Page 117: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/117.jpg)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
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
€
G−1(γ kλ (θ ) )
a map from a “disk” to itself.
So consider the composition
![Page 118: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/118.jpg)
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are needed to see this picture.
€
γ0€
vλ
€
−vλ
€
γk
€
γkλ (θ )
€
G−1
€
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 the centers of the S-holes and M-sets.€
G−1(γ kλ (θ ) )
€
γkλ (θ )
![Page 119: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/119.jpg)
Remarks: 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 120: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/120.jpg)
Part 3: Behavior of the Julia sets
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n = 2: When , the Julia set of is the unit circle. But, as , the Julia set of converges to the closed unit disk
€
Fλ
€
λ → 0
€
λ =0
€
Fλ
Part 3: Behavior of the Julia sets
![Page 122: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/122.jpg)
n > 2: J is always a Cantor set of “circles” when is small.
n = 2: When , the Julia set of is the unit circle. But, as , the Julia set of converges to the closed unit disk
€
Fλ
€
λ → 0
€
λ =0
€
Fλ
€
λ
Part 3: Behavior of the Julia sets
![Page 123: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/123.jpg)
n > 2: J is always a Cantor set of “circles” when is small.
n = 2: When , the Julia set of is the unit circle. But, as , the Julia set of converges to the closed unit disk
€
Fλ
€
λ → 0
€
λ =0
€
Fλ
Moreover, there is a such that there is always a“round” annulus of “thickness” between two of these circles in the Fatou set. So J does not converge to the unit disk when n > 2.
€
λ
€
δ > 0
€
δ
Part 3: Behavior of the Julia sets
![Page 124: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/124.jpg)
n = 2
Theorem: When n = 2, the Julia sets converge to the unit disk as
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
€
λ → 0
![Page 125: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/125.jpg)
Suppose the Julia sets do not converge to the unit disk D as
€
λ → 0
Sketch of the proof:
![Page 126: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/126.jpg)
Suppose the Julia sets do not converge to the unit disk D as
€
λ → 0
Sketch of the proof:
Then there exists and a sequence such that, for each i,there is a point such that lies in the Fatou set.
€
Bδ (zi )
€
δ > 0
€
λi → 0
€
zi ∈D
![Page 127: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/127.jpg)
Suppose the Julia sets do not converge to the unit disk D as
€
λ → 0
Sketch of the proof:
Then there exists and a sequence such that, for each i,there is a point such that lies in the Fatou set.
€
Bδ (zi )
€
λi → 0
€
zi ∈D
€
Bδ (z1)
€
Bδ (z2 )
€
Bδ (z3)
€
Bδ (z4 )
€
δ > 0
![Page 128: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/128.jpg)
Suppose the Julia sets do not converge to the unit disk D as
€
λ → 0
Sketch of the proof:
Then there exists and a sequence such that, for each i,there is a point such that lies in the Fatou set.
€
Bδ (zi )
€
λi → 0
€
zi ∈D
The must accumulate on some nonzero point, say ,so we may assume that lies in the Fatou set for all i.
€
zi
€
z*
€
Bδ (z*)
€
Bδ (z1)
€
Bδ (z2 )
€
Bδ (z3)
€
Bδ (z4 )
€
δ > 0
![Page 129: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/129.jpg)
Suppose the Julia sets do not converge to the unit disk D as
€
λ → 0
Sketch of the proof:
Then there exists and a sequence such that, for each i,there is a point such that lies in the Fatou set.
€
Bδ (zi )
€
λi → 0
€
zi ∈D
€
Bδ (z*)
The must accumulate on some nonzero point, say ,so we may assume that lies in the Fatou set for all i.
€
zi
€
z*
€
Bδ (z*)
€
δ > 0
![Page 130: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/130.jpg)
Suppose the Julia sets do not converge to the unit disk D as
€
λ → 0
Sketch of the proof:
Then there exists and a sequence such that, for each i,there is a point such that lies in the Fatou set.
€
Bδ (zi )
€
λi → 0
€
zi ∈D
€
Bδ (z*)
The must accumulate on some nonzero point, say ,so we may assume that lies in the Fatou set for all i.
€
zi
€
z*
€
Bδ (z*)
But for large i, so stretchesinto an “annulus” that surrounds the origin, so thisdisconnects the Julia set.€
Fλ i ≈ z2
€
Fλ ik
€
Bδ (z*)€
Fλk
€
δ > 0
![Page 131: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/131.jpg)
So the Fatou components must become arbitrarily small:
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€
λ =−0.0001
€
λ =−0.0000001
![Page 132: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/132.jpg)
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€
λ =.01
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€
λ =.0001
€
λ =.000001
n > 2: Note the “round” annuli in the Fatou set; there is alwayssuch an annulus of some fixed width for small.
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€
λ =.00000001
€
λ =.0000000001
€
| λ |
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€
λ =.000000000001
![Page 133: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/133.jpg)
T
B
€
| λ | small
€
⇒ T is tiny
A0
mod A0 = m is huge
Say n = 3:
![Page 134: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/134.jpg)
T
B
€
| λ | small
€
⇒ T is tiny
A0
mod A0 = m is hugeand the boundary of A0
is very close to S1
Say n = 3:
![Page 135: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/135.jpg)
T
B
€
| λ | small
€
⇒ T is tinymod A0 = m is hugeand the boundary of A0
is very close to S1
Say n = 3:
X1
A1
A1 and A1 mapped to A0;X1 is mapped to T
X1
A1
A1
~~
A0
mod A1 = mod A1 = mod X1 = m/3; A1 S1
€
⇒
€
≈
€
∂~
![Page 136: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/136.jpg)
T
B
€
| λ | small
€
⇒ T is tinymod A0 = m is hugeand the boundary of A0
is very close to S1
Say n = 3:
X1
A1
A1 and A1 mapped to A0;X1 is mapped to T
X2
A1
A2
A2 is mapped to A1;X2 is mapped to X1
mod A2 = mod X2 = m/32; A2 S1
€
⇒mod A1 = mod A1 = mod X1 = m/3; A1 S1
€
⇒
€
≈
€
∂
€
≈
€
∂
~
~
![Page 137: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/137.jpg)
B
€
| λ | small
€
⇒ T is tinymod A0 = m is hugeand the boundary of A0
is very close to S1
Say n = 3:
A1 and A1 mapped to A0;X1 is mapped to T
Xk
Ak
Ak-1
€
M
Ak is mapped to Ak-1; Xk to Xk-1
mod A2 = mod X2 = m/32; A2 S1
€
⇒mod A1 = mod A1 = mod X1 = m/3; A1 S1
~
€
⇒
€
≈
€
∂
€
≈
€
∂mod Ak = mod Xk = m/3k; Ak S1
€
⇒
€
≈
€
∂
A2 is mapped to A1;X2 is mapped to X1
~
€
N
![Page 138: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/138.jpg)
B
€
| λ | small
€
⇒ T is tinymod A0 = m is hugeand the boundary of A0
is very close to S1
Say n = 3:
Xk
Ak
Ak-1
1 mod Ak < 3
€
≤Eventually find k sothatand Ak S1
€
≈
€
∂
€
=α
![Page 139: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/139.jpg)
B
€
| λ | small
€
⇒ T is tinymod A0 = m is hugeand the boundary of A0
is very close to S1
Say n = 3:
Xk
Ak
Ak-1
1 mod Ak < 3
€
≤Eventually find k sothatand Ak S1
€
≈
€
∂
€
⇒ Ak must contain a round annulus of modulus
(Ble, Douady, and Henriksen)
€
=α
€
α −1/2 >1/2
![Page 140: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/140.jpg)
B
€
| λ | small
€
⇒ T is tinymod A0 = m is hugeand the boundary of A0
is very close to S1
Say n = 3:
Xk
Ak
Ak-1
€
⇒ Ak must contain a round annulus of modulus
(Ble, Douady, and Henriksen)
But does this annulus have definite “thickness?”
1 mod Ak < 3
€
≤Eventually find k sothatand Ak S1
€
≈
€
∂
€
=α
€
α −1/2 >1/2
![Page 141: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/141.jpg)
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€
∂Ak in here
![Page 142: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/142.jpg)
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€
∂Ak in here
€
=αmod Ak
says that the innerboundary of Ak
cannot be insideor outside ,so the round annulusin Ak is “thick”€
γ0
€
γ1
€
γ1
€
γ0
Ak
![Page 143: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/143.jpg)
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€
∂Ak in here
€
μ1
€
μ0
Ak
Same argumentsays that Ak Xk
is twice as thick
€
∪
€
∪ Xk
![Page 144: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/144.jpg)
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Xk
So Xk musthave definite thickness as well
![Page 145: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/145.jpg)
Part 4: A major application
![Page 146: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/146.jpg)
Here’s the parameter plane when n = 2:
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Qu
ickTim
e™
an
d a
TIF
F (
LZ
W)
decom
pre
sso
rare
nee
de
d t
o s
ee t
his
pic
ture
.
Rotate it by 90 degrees:
and this object appears everywhere.....
![Page 148: Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: (why the case n = 2 is )](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e7e5503460f94b8183b/html5/thumbnails/148.jpg)
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