structures in the parameter planes dynamics of the family of complex maps paul blanchard toni garijo...

Structures in the Parameter Planes

Dynamics of the family of complex maps

Paul BlanchardToni GarijoMatt HolzerU. HoomiforgotDan LookSebastian Marotta Monica Moreno RochaElizabeth RussellYakov ShapiroDavid Uminsky

with:

€

Fλ (z) = z n +λ

z n

First a brief advertisement:

AIMS Conference on Dynamical Systems, Differential Equations and Applications

Dresden University of TechnologyDresden, GermanyMay 25-28 2010

Organizers: Janina Kotus, Xavier Jarque, me

One half hour slots for speakers. Interested in attending/speaking?

Contact me at [email protected]

Structures in the Parameter Planes

What you see in the dynamical plane often reappears(enchantingly so) in the parameter plane....

Dynamics of the family of complex maps

€

Fλ (z) = z n +λ

z n

Cantor Necklaces:

A Cantor necklace in aJulia set when n = 2

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Cantor Necklaces:



A Cantor necklace in aJulia set when n = 2



and in the parameter plane



Mandelpinski Necklaces:

Circles of preimages ofthe trap door and critical

points around 0






Circles through centers ofSierpinski holes and baby M -sets in the param-plane

Circles of preimages ofthe trap door and critical

points around 0




Circles of pre-preimages ofthe trap door and pre-critical

points around 0



* the only exception

Circles through centers ofSierpinski holes and baby M*-sets in the param-plane







points around 0

* the only exception

Circles through centers ofSierpinski holes and baby M*-sets in the param-plane





points around 0







Circles through Sierpinskiholes and baby Mandelbrotsets in the parameter plane




points around 0

As Douady often said “You sow the seeds in thedynamical plane and reap the harvest in the

parameter plane.”

It is often easy to prove something in the dynamicalplane, but harder to reproduce it in the parameter plane.

Here is how we will do this:

Suppose you have some object in the dynamicalplane that varies analytically with the parametermaybe a closed curve, maybe a Cantor necklace, or:

€

λ

dynamical plane


Maybe it’s your face,so call it Face( )

€

λ

dynamical plane

€

λ


Maybe it’s your face,so call it Face( )Change , and Face( )moves analytically:

€

λ

€

λ

€

λ

dynamical plane

€

λ


Maybe it’s your face,so call it Face( )Change , and Face( )moves analytically:maybe like this

€

λ

€

λ

€

λ

dynamical plane

€

λ


Maybe it’s your face,so call it Face( )Change , and Face( )moves analytically:or like this (you’re socute!)

€

λ

€

λ

€

λ

dynamical plane

€

λ

So any particular point in Face( ), say the tip ofyour nose, nose( ), varies analytically with

€

λ

€

λ

€

λ

dynamical plane

nose( )

€

λ

So any particular point in Face( ), say the tip ofyour nose, nose( ), varies analytically with

€

λ

€

λ

dynamical plane

nose( )

€

λ€

λ

So we have an analytic function nose( ) fromparameter plane to the dynamical plane

€

λ

dynamical plane

nose( )

€

λ

parameter plane

So we have an analytic function nose( ) fromparameter plane to the dynamical plane

€

λ

€

λ

dynamical plane

nose( )

€

λ

parameter plane

Now suppose we have another analytic function G( ) taking parameter plane to dynamical plane one-to-one

€

λ

dynamical plane

nose( )

€

λ

parameter plane

€

λ

G

So we have an inverse map G-1 taking the dynamicalplane back to the parameter plane

€

λ

dynamical plane

nose( )

€

λ

parameter plane

G-1

Now suppose G takes a compact disk D in the parameterplane to a disk in dynamical plane, and nose( ) is alwayscontained strictly inside G(D) when .

€

λ

dynamical plane

nose( )

€

λ

parameter plane

G-1

€

λ

D G(D)

€

λ ∈D

So G-1(nose( )) maps D strictly inside itself.

€

λ

dynamical plane

nose( )

€

λ

parameter plane

G-1

€

λ

D G(D)

So G-1(nose( )) maps D strictly inside itself. So by the Schwarz Lemma, there is a unique fixed point for the map G-1(nose( )).

€

λ*

dynamical planeparameter plane

G-1

€

λ

D G(D)

€

λ

€

λ*

nose( )

€

λ*

is the unique parameter for which G( ) = nose( ).

€

λ*


G-1D G(D)

€

λ*

€

λ*

nose( )

€

λ*€

λ*



DG(D)

€

λ*

€

λ*

€

λ*

If we do this for each point in Face( ), we then get the same “object” in the parameter plane.

€

λ



DG(D)

€

λ*

€

λ*

€

λ*

If we do this for each point in Face( ), we then get the same “object” in the parameter plane.

€

λ

Why are you so unhappyliving in the parameter plane?

The goal today is to show the existence in theparameter plane of:

1. Cantor necklaces 2. Cantor webs 3. Mandelpinski necklaces 4. Cantor sets of circles of Sierpinski curve Julia sets

1. Cantor Necklaces

A Cantor necklace is the Cantor middle thirds set with open disks replacing the removed intervals.

1. Cantor Necklaces




a Julia set with n = 2 anda Cantor necklace

1. Cantor Necklaces


a Julia set with n = 2 andanother Cantor necklace



1. Cantor Necklaces


a Julia set with n = 2 andlots of Cantor necklaces



And there are Cantor necklaces in the parameter planes.



n = 2

1. Cantor Necklaces



n = 2

1. Cantor Necklaces

We’ll just show the existence of this Cantor necklacealong the negative real axis when n = 2.



n = 2

1. Cantor Necklaces

€

Fλ (z ) = z 3 +λ

z 3



Recall:

B = immediate basin of T = trap door

€

∞

B

T

2n free critical points

€

cλ = λ1/2n€

Fλ (z ) = z 3 +λ

z 3



cc

Recall:


€

∞

B

T


€

cλ = λ1/2n

Only 2 critical values

€

vλ = ±2 λ

€

Fλ (z ) = z 3 +λ

z 3



cc

v

Recall:


€

∞

B

T


€

cλ = λ1/2n

Only 2 critical values

€

vλ = ±2 λ

€

Fλ (z ) = z 3 +λ

z 3

2n prepoles

€

pλ = (−λ )1/2n



cc

v

p

p

Recall:


€

∞

B

0

Consider

€

Fλ (z) = z2 +λ

z2 , λ < 0

Since , preserves the real line.

Consider

€

Fλ (z) = z2 +λ

z2 , λ < 0

€

Fλ

€

λ < 0

Consider

€

Fλ (z) = z2 +λ

z2 , λ < 0

graph of

€

Fλ (x)

need a glassof wine???

Consider

€

Fλ (z) = z2 +λ

z2 , λ < 0

B = basin of infinity

graph of

€

Fλ (x)

Consider

€

Fλ (z) = z2 +λ

z2 , λ < 0


T = trap door

graph of

€

Fλ (x)

Consider

€

Fλ (z) = z2 +λ

z2 , λ < 0


T = trap door

graph of

€

Fλ (x)

I0

I1The two intervals I0 andI1 are expanded over theunion of these intervalsand the trap door.

Consider

€

Fλ (z) = z2 +λ

z2 , λ < 0


T = trap door

graph of

€

Fλ (x)

I0

So there is an invariant Cantorset on the negative real axis.

The two intervals I0 andI1 are expanded over theunion of these intervalsand the trap door.

I1

Consider

€

Fλ (z) = z2 +λ

z2 , λ < 0


T = trap door

graph of

€

Fλ (x)

I0

So there is an invariant Cantorset on the negative real axis.Add in the preimages of Tto get the Cantor necklace inthe dynamical plane for .

The two intervals I0 andI1 are expanded over theunion of these intervalsand the trap door.

I1

€

λ < 0



The Cantor necklace for negative

€

λ



This portion is also a Cantor necklacelying on the negative real axis for

€

λ < 0

And we have a similar Cantor necklacelying on the negative real axis in theparameter plane for n = 2.



To see this, let D be the half-disk |z| < 1, Re(z) < 0.

D

Let be the second iterate of the critical point

€

G(λ ) = Fλ (vλ )

D

€

G(λ )



€

G(λ ) = Fλ (vλ ) = (2 λ )2 +λ

(2 λ )2

D

€

G(λ )



€

G(λ ) = Fλ (vλ ) = (2 λ )2 +λ

(2 λ )2= 4λ +

1

4

D

€

G(λ )



So G is 1-to-1 on D,and maps D over itself;

D

G

€

G(λ )

€

G(λ ) = Fλ (vλ ) = (2 λ )2 +λ

(2 λ )2= 4λ +

1

4

.25-3.75



So G is 1-to-1 on D,and maps D over itself;equivalently, G-1 contractsG(D) inside itself. D

G-1

€

G(λ )

€

G(λ ) = Fλ (vλ ) = (2 λ )2 +λ

(2 λ )2= 4λ +

1

4

.25-3.75


For any in D (not just ),we also have an invariant Cantor set as we showed earlier:

€

λ

€

λ < 0

U2

U0

€

cλ


€

λ

€

λ < 0

U0 and U2 are portionsof a prepole sector

U2

U0

€

cλ


€

λ

€

λ < 0

U0 and U2 are portionsof a prepole sectorthat are each mappedunivalently over bothU0 and U2.

U0

€

cλ


So there is a portion of a Cantor set lying in U2.


€

λ

€

λ < 0

U0


€

λ


€

cλ


And we can add in theappropriate preimages ofthe trap door to get aCantor necklace.

€

λ < 0

And, since lies in D, theCantor set lies inside G(D).

€

λ

G(D)


€

λ

€

λ < 0



And we can add in theappropriate preimages ofthe trap door to get aCantor necklace.

U0

U2

We can identify each point in the Cantor set in U2 bya unique sequence of 0’s and 2’s: s = (2 s1 s2 s3 ....)given by the itinerary of the point.

So, for each such sequence s,we have a map , which is defined on D anddepends analytically on

€

λ → zs (λ )

€

λ

We can identify each point in the Cantor set in U2 bya unique sequence of 0’s and 2’s: s = (2 s1 s2 s3 ....)given by the itinerary of the point.

U0

U2€

zs (λ )

We therefore have two maps defined on D:

D

G(D)


1. The univalent map

€

G(λ ) = 4λ +1

4

D

G G(D)


1. The univalent map

€

G(λ ) = 4λ +1

4

D

G

2. The point in the Cantor set

€

zs (λ )

€

zs

G(D)

D

G-1

€

G−1 ozs

€

zs

G(D)

So maps D strictly inside itself;

D

G-1

€

G−1 ozs

€

zs

G(D)

So maps D strictly inside itself; bythe Schwarz Lemma, there is a unique fixed point in D for this map.

€

λs*

€

λs*

For this parameter, we have , sothis is the unique parameter for which the criticalorbit lands on the point .

€

G(λ s* ) = zs (λ s

* )

D

G-1

€

G−1 ozs

€

zs

G(D)€

zs (λ )

So maps D strictly inside itself; bythe Schwarz Lemma, there is a unique fixed point in D for this map.

€

λs*

D

€

λs*

Claim: this Cantor set lies on the negative real axis.

This produces a Cantor set of parameters ,one for each sequence s.

€

λs*

€

G(λ ) = 4λ +1/ 4

Recall:

, so G decreases from .25 to -3.75 as goes from 0 to -1 in D.

€

λ



€

λs*

, so G decreases from .25 to -3.75 as goes from 0 to -1 in D.

Recall:


the Cantor set in the dynamical planelies on the negative real axis when .

€

λ < 0


€

λs*

€

G(λ ) = 4λ +1/ 4

€

λ

Recall:

So must hit each point in the Cantor set alongthe negative axis at least once.

€

G(λ )



€

λ < 0


€

λs*

€

G(λ ) = 4λ +1/ 4 , so G decreases from .25 to -3.75 as goes from 0 to -1 in D.

€

λ

Recall:

So must hit each point in the Cantor set alongthe negative axis at least once.

€

G(λ )

So each parameter in the parameter plane necklacemust also lie in [-1, 0]. This produces the Cantor set portion of the necklace on the negative real axis.

€

λs*



€

λ < 0


€

λs*

€

G(λ ) = 4λ +1/ 4 , so G decreases from .25 to -3.75 as goes from 0 to -1 in D.

€

λ

Similar arguments produce parameters onthe negative axis that land after a specifieditinerary on a particular point in B (thatis determined by the Böttcher coordinate).

Similar arguments produce parameters onthe negative axis that land after a specifieditinerary on a particular point in B (thatis determined by the Böttcher coordinate).And then these intervals can be expandedto get the Sierpinski holes in the necklace.



2. Cantor webs





n = 4

Recall that, when n > 2, we have Cantor “webs” in the dynamical plane:

n = 3





2. Cantor webs

Recall that, when n > 2, we have Cantor “webs” in the dynamical plane:

n = 3n = 3





2. Cantor webs

When n > 2, we also have Cantor “webs” in the parameter plane:

n = 4n = 3

A slightly different argument as in the case of theCantor necklaces works here. Say n = 3.



n = 3

U1

U2

U4

U5€

vλ

€

−vλ

In the dynamical plane, wehad the disks Uj.



n = 3

U1

U2

U4

U5€

vλ

€

−vλ

Each of these Uj were mappedunivalently over all the others,

excluding U0 and Un, sowe found an invariant Cantor

set in these regions.


U0

U3




n = 3

U1

U2

U4

U5€

vλ

€

−vλ

Each of these Uj were mappedunivalently over all the others,

excluding U0 and U3, sowe found an invariant Cantor

set in these regions.


U0

U3

U0 and U3 are mappedunivalently over these Uj,

so there is a preimage ofthis Cantor set in both U0

and U3


Now let be one of the two critical values, so

€

G(λ )

€

G(λ ) = 2 λ



U2

U4

€

−vλ

U3

D

And choose a disk D in one of the “symmetry sectors”

in the parameter plane:


€

G(λ )

€

G(λ ) = 2 λ





U1

U2

U4

U5€

vλ

€

−vλ

U0

U3

D



Then G maps D univalentlyover all of U0, so we again get a copy of the Cantor set in D

G


€

G(λ )

€

G(λ ) = 2 λ





U1

U2

U4

U5€

vλ

€

−vλ

U0

U3

D



Then G maps D univalentlyover all of U0, so we again get a copy of the Cantor set in D

G

Then adjoining the appropriate Sierpinski holesgives a Cantor web in the parameter plane.

3. “Mandelpinski” necklaces

A Mandlepinski necklace is a simple closed curve inthe parameter plane that passes alternately through acertain number of centers of baby M-sets and thesame number of centers of S-holes.

oops, sorry....

A Mandlepinski necklace is a simple closed curve inthe parameter plane that passes alternately through acertain number of centers of baby M-sets and thesame number of centers of S-holes.


A Mandlepinski necklace is a simple closed curve inthe parameter plane that passes alternately through acertain number of centers of baby M-sets and thesame number of centers of Sierpinski-holes.




A Julia set



parameter plane n = 4






There is a “ring” around Tpassing through 8 = 2*4

preimages of T







parameter plane n = 4 There is a “ring” around Tpassing through 8 = 2*4

preimages of T




Another “ring” around Tpassing through 32 = 2*42

preimages of T







Another “ring” around Tpassing through 128 = 2*43

preimages of T





parameter plane for n = 4

Now look around the McMullen domain in the parameter plane:






There is a ring around M that passes alternately through the centers of

3 = 2*40 + 1 Sierpinski holes and 3 Mandelbrot sets

Another ring around M that passes alternately through the centers of

9 = 2*41 + 1 Sierpinski holes and 9 “Mandelbrot sets”*




*well, 3 period 2 bulbs

Then 33 = 2*42 + 1 Sierpinski holes and 33 Mandelbrot sets




Then 129 = 2*43 + 1 Sierpinski holes and 129 Mandelbrot sets




parameter plane for n = 3



Similar kinds of rings occur in the other parameter planes:

n = 3



Similar kinds of rings occur in the other parameter planes:

S0: 2 = 1*30 + 1 Sierpinski holes & M-sets

S0

n = 3

S1: 4 = 1*31 + 1 Sierpinski holes & M-sets*

*well, two period 2 bulbs

n = 3



S2: 10 = 1*32 + 1 Sierpinski holes & “M-sets”



n = 3

S3: 28 = 1*33 + 1 Sierpinski holes & M-sets



n = 3

82, 244, then 730 Sierpinski holes...



n = 3

the 13th ring passes through1,594,324 Sierpinski holes...



n = 3

sorry, I forgot.....nevermind

* with one exception

Theorem: For each n > 2, the McMullen domain issurrounded by infinitely many simple closed curves Sk

(“Mandelpinski” necklaces) having the property that:1. each Sk surrounds the McMullen domain and Sk+1, and the Sk accumulate on the boundary of M;

• Sk meets the center of exactly (n-2)nk-1 + 1 Sierpinski holes, each with escape time k + 2;1. Sk also passes through the centers of the same number of baby Mandelbrot sets*

The critical points andprepoles all lie on the “critical circle”

€

r = | λ |1/2n pc

p

c


The critical circle is mapped 2n-to-1 ontothe “critical value ray”

v

0

pc

p

c

€

r = | λ |1/2n



v

0

And every other circle centeredat the origin and outside thecritical circle is mapped n-to-1to an ellipse with foci at the criticalvalues

€

r = | λ |1/2n



v

0

And every other circle centeredat the origin and outside thecritical circle is mapped n-to-1to an ellipse with foci at the criticalvalues, and same inside

€

r = | λ |1/2n

There are no critical pointsoutside the critical circle, so this region is mapped asn-to-1 covering onto thecomplement of thecritical value ray.

v

0

v

0 The interior of thecritical circle is also mapped n-to-1 onto thecomplement of thecritical value ray

There are no critical pointsoutside the critical circle, so this region is mapped asn-to-1 covering onto thecomplement of thecritical value ray.

The dividing circle contains all parameters for whichthe critical values lie on the critical circle, i.e.,

€

| λ |1/2n= 2 | λ |1/2 ⇒ | λ | = 2−2n /(n−1)

The dividing circle contains all parameters for whichthe critical values lie on the critical circle, i.e.,

€

| λ |1/2n= 2 | λ |1/2 ⇒ | λ | = 2−2n /(n−1)



When n = 4, the dividing circle passes through 3

centers of Sierpinski holes and 3 baby Mandelbrot sets

€

n = 4 : | λ | = 2−8/3

The dividing circle passes through n-1 centers of Sierpinski holes and n-1 centers of baby Mandelbrot sets.



When n = 4, the dividing circle passes through 3

centers of Sierpinski holes and 3 baby Mandelbrot sets

€

n = 4 : | λ | = 2−8/3

Reason:

€

r = | λ |8€

| λ | = 2−8/3

parameter plane n = 4 €

vλ

€

λ

dynamical plane

Reason: as runs once around the dividing circle,

€

r = | λ |8€

| λ | = 2−8/3

€

λ

parameter plane n = 4 €

vλ

€

λ

dynamical plane

€

r = | λ |8

€

vλ

€

| λ | = 2−8/3€

λ

Reason: as runs once around the dividing circle, rotates 1/2 of a turn,

€

λ

€

vλ


dynamical plane

Reason: as runs once around the dividing circle, rotates 1/2 of a turn, while the critical points and prepoles

each rotate on 1/8 of a turn.

€

r = | λ |8

€

λ

€

vλ

€

| λ | = 2−8/3€

λ


dynamical plane

€

vλ

Reason: as runs once around the dividing circle, rotates 1/2 of a turn, while the critical points and prepoles

each rotate on 1/8 of a turn.

€

r = | λ |8

€

λ

€

vλ

€

vλ

So meets 3 prepoles and critical points enroute.

€

vλ

€

| λ | = 2−8/3€

λ


dynamical plane



So the ring S0 is just the dividing circle in parameter plane.

S0

n = 4



So the ring S0 is just the dividing circle in parameter plane.

S0

For the other rings, let’s consider for simplicity only the case where n = 4

n = 4

When lies inside the dividing circle, we have

€

λ

€

| vλ | = 2 | | λ | < | λ |1/8 = | cλ |

When lies inside the dividing circle, we have

€

λ

so maps the critical circle C0 strictly inside itself

€

Fλ

€

vλ

€

−vλ C0

€

| vλ | = 2 | | λ | < | λ |1/8 = | cλ |

€

vλ

€

−vλ

Now there is a preimage C1 of the critical circle thatis mapped 4-to-1 onto the critical circle, and this curvecontains 32 pre-critical points and 32 pre-pre-poles.

C0

C1

€

vλ

€

−vλ

And then a preimage C2 of the C1 that is mapped 4-to-1 onto the C1, and so 16-to-1 onto C0, and this curve contains 128 pre-pre-critical points and 128 pre-pre-pre-poles, etc.

C0

C1

C2



The rings C0 and C1


€

G(λ ) = Fλ2 (cλ ) = 2nλn /2 +

1

2nλ (n /2)−1

€

G(λ )


€

G(λ ) = 16λ2 +1

16λSo when n = 4.

€

G(λ ) = Fλ2 (cλ ) = 2nλn /2 +

1

2nλ (n /2)−1

€

G(λ )


€

G(λ ) = 16λ2 +1

16λSo when n = 4.

€

G(λ ) = Fλ2 (cλ ) = 2nλn /2 +

1

2nλ (n /2)−1

€

G(λ )

Note that

€

G(λ ) → ∞ as

€

λ → 0 provided n > 2.

When n = 2,

€

G(λ ) = 4λ +1

4, a very different situation.



G maps points in the parameter plane to points in the dynamical plane

C0

G

the critical circle



G

Let D be the open disk of radius 1/8 in the parameter plane.G maps D univalently onto a region in the exterior of C0

G(D)

C0

D



and G(D) covers C1, C2,...

C0

D

G



Choose a small disk D0 inside M. Then G maps the annulus A = D - D0 univalently over all of the Cj, j > 0.

C0

D0

A

G



Choose a parametrization of Ck, say . So wehave a second map from A into G(A),

C0

D0

A

€

λ → γλk (θ )

€

γλk (θ )

€

γλk (θ )

G



Since G is 1-to-1, we thus have a map which takes A into A.

C0

D0

A

€

H (λ ) =G−1(γ λk (θ ))

€

γλk (θ )

H



C0

D0

A

€

γλk (θ )

H

Let S be the covering strip of A and let H*: S S be the covering map of H: A A


By the Schwarz Lemma, for each given k, , and , there is a unique fixed point for H* in A, which depends analytically on .

€

ηλk (θ )

€

θ

€

λ

€

λ


By the Schwarz Lemma, for each given k, , and , there is a unique fixed point for H* in A, which depends analytically on .

€

ηλk (θ )

€

θ

€

λ

So the map gives a parametrization of thering Sk in the parameter plane, and -values that

correspond to pre-poles or pre-critical points are thenparameters at the centers of Sierpinski holes or baby

Mandelbrot sets.

€

λ

€

λ → ηλk (θ )

€

θ

There are (n - 2)nk-1 pre-poles in the kth dynamical planering, but (n - 2)nk-1 + 1 centers of Sierpinski holes inthe parameter plane rings. Here’s the reason:

On the annulus A,

€

G(λ ) = 16λ2 +1

16λ≈

1

16λ


On the annulus A,

€

G(λ ) = 16λ2 +1

16λ≈

1

16λ

So as rotates clockwise around the ring Sk, rotates once around the origin in the counterclockwise direction. Meanwhile, each pre-pole and pre-critical pointrotates clockwise by approximately 1/((n-2)nk-1 of a turn.

€

G(λ )

€

λ


On the annulus A,

€

G(λ ) = 16λ2 +1

16λ≈

1

16λ

So hits one additional prepole or pre-critical pointwhile traveling around each Sk.

€

G(λ )


So as rotates clockwise around the ring Sk, rotates once around the origin in the counterclockwise direction. Meanwhile, each pre-pole and pre-critical pointrotates clockwise by approximately 1/((n-2)nk-1 of a turn.

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G(λ )

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λ

Similar arguments show that each Sierpinski holeon a Mandelpinski necklace is also surroundedby infinitely many sub-Mandelpinski necklaces



Some open problems:

Some open problems:

1. By Xiaoguang Wang’s result yesterday, the boundary of B is always a simple closed curve (except when J is a Cantor set) when n > 2.



n = 3

Some open problems:

1. By Xiaoguang Wang’s result yesterday, the boundary of B is always a simple closed curve (except when J is a Cantor set) when n > 2. Is the boundary of the parameter plane also a simple closed curve???





n = 3 n = 3

Some open problems:


n = 4 n = 4





Some open problems:


2. What about the crazy case n = 2???

Some open problems:



3. Are the Julia sets for these maps always locally connected?

Some open problems:



3. Are the Julia sets for these maps always locally connected?

4. Are the parameter planes locally connected???

5. What is going on in the parameter plane near 0 when n = 2?



6. What is the structure in the parameter plane outside the dividing circle?





7. What is going on in the parameter plane for the maps

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Fλ (z) = zn +λ

z



n = 2, d = 1

Not a babyM-set

€

Fλ (z) = zn +λ

z



n = 2, d = 1

No Cantornecklace


Not a babyM-set



€

Fλ (z) = zn +λ

z

n = 2, d = 1

No Cantornecklace


€

Fλ (z) = zn +λ

z

n = 4, d = 1




J approaches theunit disk only alongthese 3 lines

structures in the parameter planes dynamics of the family of complex maps paul blanchard toni garijo...

Documents

parameter plane slide

dynamical plane slide

paramplane slide

cantor necklaces

mandelpinski necklaces

precritical points

exception circles

trap door