filled julia sets of complex quadratic maps · 2019. 4. 2. · filled julia sets of complex...

Filled Julia sets of complex quadratic maps

c = −2.5

I Points further than

√14 + d(c , 0)

from 0 always fly offtoward ∞.

I These points form aset L0.

I The points that reachL0 after n steps forma set Ln.

c = −2.5

I Cut out L0.

I Cut out L1.

I Cut out L2.

I Cut out L3.

I Cut out L4.

I Cut out L5.

I Cut out L6.

I Cut out L7.

c = −2.5

I Cut out L0.

I Cut out L1.

I Cut out L2.

I Cut out L3.

I Cut out L4.

I Cut out L5.

I Cut out L6.

I Cut out L7.

c = −2.5

I Cut out L0.

I Cut out L1.

I Cut out L2.

I Cut out L3.

I Cut out L4.

I Cut out L5.

I Cut out L6.

I Cut out L7.

c = −2.5

I Cut out L0.

I Cut out L1.

I Cut out L2.

I Cut out L3.

I Cut out L4.

I Cut out L5.

I Cut out L6.

I Cut out L7.

c = −2.5

I Cut out L0.

I Cut out L1.

I Cut out L2.

I Cut out L3.

I Cut out L4.

I Cut out L5.

I Cut out L6.

I Cut out L7.

c = −2.5

I Cut out L0.

I Cut out L1.

I Cut out L2.

I Cut out L3.

I Cut out L4.

I Cut out L5.

I Cut out L6.

I Cut out L7.

c = −2.5

I Cut out L0.

I Cut out L1.

I Cut out L2.

I Cut out L3.

I Cut out L4.

I Cut out L5.

I Cut out L6.

I Cut out L7.

c = −2.5

I Cut out L0.

I Cut out L1.

I Cut out L2.

I Cut out L3.

I Cut out L4.

I Cut out L5.

I Cut out L6.

I Cut out L7.

c = −2.5

I Cut out L0.

I Cut out L1.

I Cut out L2.

I Cut out L3.

I Cut out L4.

I Cut out L5.

I Cut out L6.

I Cut out L7.

c = −2.5

I Cut out L0.

I Cut out L1.

I Cut out L2.

I Cut out L3.

I Cut out L4.

I Cut out L5.

I Cut out L6.

I Cut out L7.

c = −2.5

I When c is in thelower region,

Qc : C→ C

has the same filledJulia set as

Qc : R→ R.

I We get a nice way tovisualize this weirdsubset of R.

c = −2

I At the very top of thelower region, the filledJulia set is

[−2, 2] ⊂ R,

as expected.

c = −2

[−2, 2] ⊂ R,

as expected.

c = −2

[−2, 2] ⊂ R,

as expected.

c = −2

[−2, 2] ⊂ R,

as expected.

c = −2

[−2, 2] ⊂ R,

as expected.

c = −2

[−2, 2] ⊂ R,

as expected.

c = −2

[−2, 2] ⊂ R,

as expected.

c = −2

[−2, 2] ⊂ R,

as expected.

c = −2

[−2, 2] ⊂ R,

as expected.

c = −2

[−2, 2] ⊂ R,

as expected.

c = −0.3

I When c is in theupper region, thefilled Julia set extendsabove and below thereal axis.

c = −0.3

c = −1

I The people who firststudied complexquadratic maps gavefancy names to filledJulia sets they liked.

I This one is called thebasilica.

I Each lobe contains aneventually periodicpoint.

c = −1

c = −0.5 + 0.5i

I The constant cdoesn’t have to bereal.

c = −0.5 + 0.5i

c = −0.512511498

387847167

+ 0.521295573

094847167i

I Tiny changes in c canhave a big effect onthe filled Julia set.

I Source: Martin Doege

I https://en.wikipedia.org/wiki/

Julia_set#/media/File:

Julia_set,_plotted_with_

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

c = −0.512511498

387847167

+ 0.521295573

094847167i

Matplotlib.svg

A tour of the zoo

c = −0.122 + 0.745i .

I Playing with c , youcan find all sorts ofcool shapes.

I This kind is called aDouady rabbit.

I Source: An Introduction

to Chaotic Dynamical

Systems, plates 3–4

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

A tour of the zoo

c = −0.122 + 0.745i .

I It’s named afterAdrien Douady, oneof the first people tostudy the dynamics ofcomplex quadraticmaps.

A tour of the zoo

c = 0.360284 + 0.100376i .

I Devaney calls thiskind is called adragon.

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = 0.360284 + 0.100376i .

A tour of the zoo

c = −0.835− 0.2321i

I This one starts outlooking nice and fat.

I But those earlyapproximations aredeceptive!

I Source: Bernardo Galvao

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

A tour of the zoo

c = −0.835− 0.2321i

de Sousa’s MAT335

notes.

Special kinds of filled Julia sets

c = −1 + 23 i

I Once we cut outL0, . . . , L3, we’re leftwith two separatepieces.

I The next cut splitseach piece in two.

I The next cut does thesame thing.

c = −1 + 23 i

I In the end, the filledJulia set breaks downinto infinitely manyseparate points.

I This kind of Julia setis called Cantor dust.

c = −1 + 23 i

I After the first split,we get two “1st-levelpieces.”

I Call them I0 and I1.

I Label the each2nd-level pieceaccording to whereit’s sent by Qc , likewe did before.

I Same for each3rd-level piece.

c = −1 + 23 i

I We can define anitinerary map

τ : Kc → 2N.

I It’s a conjugacy from

Qc : Kc → Kc .

to the shift map.

I This works wheneverKc is Cantor dust.

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

I This Julia set doesn’tquite break intoseparate pieces.

I Neighboring lobestouch at a singlepoint.

I One lobe maps toitself. It’s called theSiegel disk.

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

I On the boundary ofthe Siegel disk, Qc isconjugate to Rθ.

I The lobe labeled nmaps to the Siegeldisk after n steps.

I I’ll this kind of Juliaset a “Siegel cactus.”

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = 2π(√

5/3− 1)

c = e iθ

2 (1− e iθ

θ = π(3−√

I Here’s another Siegelcactus.

I The formula for cshown above makesKc a Siegel cactuswhenever θ

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

c = e iθ

2 (1− e iθ

θ = π(3−√

2π isirrational.

filled julia sets of complex quadratic maps · 2019. 4. 2. · filled julia sets of complex...

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