Structures in the Parameter Planes Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta Monica Moreno Rocha Elizabeth Russell Yakov Shapiro David Uminsky with:

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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

Cantor Necklaces: A Cantor necklace in a Julia set when n = 2

Cantor Necklaces: A Cantor necklace in a Julia set when n = 2 and in the parameter plane

Cantor Necklaces: A Cantor necklace in a Julia set when n = 2 and in the parameter plane

Cantor Necklaces: A Cantor necklace in a Julia set when n = 2 and in the parameter plane

Mandelpinski Necklaces: Circles of preimages of the trap door and critical points around 0

Mandelpinski Necklaces: Circles through centers of Sierpinski holes and baby M -sets in the param-plane Circles of preimages of the trap door and critical points around 0

Mandelpinski Necklaces: Circles of pre-preimages of the trap door and pre-critical points around 0 * the only exception Circles through centers of Sierpinski holes and baby M*-sets in the param-plane

Mandelpinski Necklaces: Circles of pre-preimages of the trap door and pre-critical points around 0 * the only exception Circles through centers of Sierpinski holes and baby M*-sets in the param-plane

Mandelpinski Necklaces: Circles of pre-preimages of the trap door and pre-critical points around 0 Circles through centers of Sierpinski holes and baby M -sets in the param-plane

Mandelpinski Necklaces: Circles of pre-preimages of the trap door and pre-critical points around 0 Circles through centers of Sierpinski holes and baby M -sets in the param-plane

Mandelpinski Necklaces: Circles of pre-preimages of the trap door and pre-critical points around 0 Circles through centers of Sierpinski holes and baby M -sets in the param-plane

Mandelpinski Necklaces: Circles through Sierpinski holes and baby Mandelbrot sets in the parameter plane Circles of pre-preimages of the trap door and pre-critical points around 0

As Douady often said “You sow the seeds in the dynamical plane and reap the harvest in the parameter plane.” It is often easy to prove something in the dynamical plane, but harder to reproduce it in the parameter plane. Here is how we will do this:

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

Suppose you have some object in the dynamical plane that varies analytically with the parameter maybe a closed curve, maybe a Cantor necklace, or: Maybe it’s your face, so call it Face( ) dynamical plane

Suppose you have some object in the dynamical plane that varies analytically with the parameter maybe a closed curve, maybe a Cantor necklace, or: Maybe it’s your face, so call it Face( ) Change, and Face( ) moves analytically: dynamical plane

Suppose you have some object in the dynamical plane that varies analytically with the parameter maybe a closed curve, maybe a Cantor necklace, or: Maybe it’s your face, so call it Face( ) Change, and Face( ) moves analytically: maybe like this dynamical plane

Suppose you have some object in the dynamical plane that varies analytically with the parameter maybe a closed curve, maybe a Cantor necklace, or: Maybe it’s your face, so call it Face( ) Change, and Face( ) moves analytically: or like this (you’re so cute!) dynamical plane

So any particular point in Face( ), say the tip of your nose, nose( ), varies analytically with dynamical plane nose( )

So any particular point in Face( ), say the tip of your nose, nose( ), varies analytically with dynamical plane nose( )

So any particular point in Face( ), say the tip of your nose, nose( ), varies analytically with dynamical plane nose( )

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

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

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

So we have an inverse map G -1 taking the dynamical plane back to the parameter plane dynamical plane nose( ) parameter plane G -1

Now suppose G takes a compact disk D in the parameter plane to a disk in dynamical plane, and nose( ) is always contained strictly inside G(D) when. dynamical plane nose( ) parameter plane G -1 D G(D)

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

is the unique parameter for which G( ) = nose( ). dynamical plane parameter plane G -1 D G(D) nose( )

is the unique parameter for which G( ) = nose( ). dynamical plane parameter plane D G(D) If we do this for each point in Face( ), we then get the same “object” in the parameter plane.

is the unique parameter for which G( ) = nose( ). dynamical plane parameter plane D G(D) If we do this for each point in Face( ), we then get the same “object” in the parameter plane. Why are you so unhappy living in the parameter plane?

The goal today is to show the existence in the parameter 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 Cantor necklace is the Cantor middle thirds set with open disks replacing the removed intervals. a Julia set with n = 2 and a Cantor necklace

1. Cantor Necklaces A Cantor necklace is the Cantor middle thirds set with open disks replacing the removed intervals. a Julia set with n = 2 and another Cantor necklace

1. Cantor Necklaces A Cantor necklace is the Cantor middle thirds set with open disks replacing the removed intervals. a Julia set with n = 2 and lots 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 necklace along the negative real axis when n = 2. n = 2 1. Cantor Necklaces

Recall: B = immediate basin of T = trap door B T

2n free critical points cc Recall: B = immediate basin of T = trap door B T

2n free critical points Only 2 critical values cc v Recall: B = immediate basin of T = trap door B T

2n free critical points Only 2 critical values 2n prepoles cc v p p Recall: B = immediate basin of T = trap door B 0

Consider

Since, preserves the real line. Consider

graph of need a glass of wine???

Consider B = basin of infinity graph of

Consider B = basin of infinity T = trap door graph of

Consider B = basin of infinity T = trap door graph of I0I0 I1I1 The two intervals I 0 and I 1 are expanded over the union of these intervals and the trap door.

Consider B = basin of infinity T = trap door graph of I0I0 So there is an invariant Cantor set on the negative real axis. The two intervals I 0 and I 1 are expanded over the union of these intervals and the trap door. I1I1

Consider B = basin of infinity T = trap door graph of I0I0 So there is an invariant Cantor set on the negative real axis. Add in the preimages of T to get the Cantor necklace in the dynamical plane for. The two intervals I 0 and I 1 are expanded over the union of these intervals and the trap door. I1I1

The Cantor necklace for negative

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

And we have a similar Cantor necklace lying on the negative real axis in the parameter 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 D To see this, let D be the half-disk |z| < 1, Re(z) < 0.

Let be the second iterate of the critical point D To see this, let D be the half-disk |z| < 1, Re(z) < 0.

Let be the second iterate of the critical point D To see this, let D be the half-disk |z| < 1, Re(z) < 0.

Let be the second iterate of the critical point So G is 1-to-1 on D, and maps D over itself; D G To see this, let D be the half-disk |z| < 1, Re(z) < 0.

Let be the second iterate of the critical point So G is 1-to-1 on D, and maps D over itself; equivalently, G -1 contracts G(D) inside itself. D G To see this, let D be the half-disk |z| < 1, Re(z) < 0.

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

U2U2 U0U0 For any in D (not just ), we also have an invariant Cantor set as we showed earlier: U 0 and U 2 are portions of a prepole sector

U2U2 U0U0 For any in D (not just ), we also have an invariant Cantor set as we showed earlier: U 0 and U 2 are portions of a prepole sector that are each mapped univalently over both U 0 and U 2.

U0U0 U 0 and U 2 are portions of a prepole sector that are each mapped univalently over both U 0 and U 2. So there is a portion of a Cantor set lying in U 2. For any in D (not just ), we also have an invariant Cantor set as we showed earlier:

U0U0 For any in D (not just ), we also have an invariant Cantor set as we showed earlier: U 0 and U 2 are portions of a prepole sector that are each mapped univalently over both U 0 and U 2. So there is a portion of a Cantor set lying in U 2. And we can add in the appropriate preimages of the trap door to get a Cantor necklace.

And, since lies in D, the Cantor set lies inside G(D). G(D) For any in D (not just ), we also have an invariant Cantor set as we showed earlier: U 0 and U 2 are portions of a prepole sector that are each mapped univalently over both U 0 and U 2. So there is a portion of a Cantor set lying in U 2. And we can add in the appropriate preimages of the trap door to get a Cantor necklace.

U0U0 U2U2 We can identify each point in the Cantor set in U 2 by a unique sequence of 0’s and 2’s: s = (2 s 1 s 2 s 3....) given by the itinerary of the point.

So, for each such sequence s, we have a map, which is defined on D and depends analytically on We can identify each point in the Cantor set in U 2 by a unique sequence of 0’s and 2’s: s = (2 s 1 s 2 s 3....) given by the itinerary of the point. U0U0 U2U2

We therefore have two maps defined on D: D G(D)

We therefore have two maps defined on D: 1. The univalent map D G G(D)

We therefore have two maps defined on D: 1. The univalent map D G 2. The point in the Cantor set G(D)

D G -1 G(D) So maps D strictly inside itself;

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

For this parameter, we have, so this is the unique parameter for which the critical orbit lands on the point. D G -1 G(D) So maps D strictly inside itself; by the Schwarz Lemma, there is a unique fixed point in D for this map. D

Claim: this Cantor set lies on the negative real axis. This produces a Cantor set of parameters, one for each sequence s.

Recall:, so G decreases from.25 to as goes from 0 to -1 in D. Claim: this Cantor set lies on the negative real axis. This produces a Cantor set of parameters, one for each sequence s.

, so G decreases from.25 to as goes from 0 to -1 in D. Recall: Claim: this Cantor set lies on the negative real axis. the Cantor set in the dynamical plane lies on the negative real axis when. This produces a Cantor set of parameters, one for each sequence s.

Recall: So must hit each point in the Cantor set along the negative axis at least once. Claim: this Cantor set lies on the negative real axis. the Cantor set in the dynamical plane lies on the negative real axis when. This produces a Cantor set of parameters, one for each sequence s., so G decreases from.25 to as goes from 0 to -1 in D.

Recall: So must hit each point in the Cantor set along the negative axis at least once. So each parameter in the parameter plane necklace must also lie in [-1, 0]. This produces the Cantor set portion of the necklace on the negative real axis. Claim: this Cantor set lies on the negative real axis. the Cantor set in the dynamical plane lies on the negative real axis when. This produces a Cantor set of parameters, one for each sequence s., so G decreases from.25 to as goes from 0 to -1 in D.

Similar arguments produce parameters on the negative axis that land after a specified itinerary on a particular point in B (that is determined by the Böttcher coordinate).

Similar arguments produce parameters on the negative axis that land after a specified itinerary on a particular point in B (that is determined by the Böttcher coordinate). And then these intervals can be expanded to 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 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 = 3

2. Cantor webs When n > 2, we also have Cantor “webs” in the parameter plane: n = 4n = 3

2. Cantor webs When n > 2, we also have Cantor “webs” in the parameter plane: n = 4n = 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 the Cantor necklaces works here. Say n = 3. n = 3 U1U1 U2U2 U4U4 U5U5 In the dynamical plane, we had the disks U j.

n = 3 U1U1 U2U2 U4U4 U5U5 Each of these U j were mapped univalently over all the others, excluding U 0 and U n, so we found an invariant Cantor set in these regions. In the dynamical plane, we had the disks U j. U0U0 U3U3 A slightly different argument as in the case of the Cantor necklaces works here. Say n = 3.

n = 3 U1U1 U2U2 U4U4 U5U5 Each of these U j were mapped univalently over all the others, excluding U 0 and U 3, so we found an invariant Cantor set in these regions. In the dynamical plane, we had the disks U j. U0U0 U3U3 U 0 and U 3 are mapped univalently over these U j, so there is a preimage of this Cantor set in both U 0 and U 3 A slightly different argument as in the case of the Cantor necklaces works here. Say n = 3.

Now let be one of the two critical values, so U2U2 U4U4 U3U3 D And choose a disk D in one of the “symmetry sectors” in the parameter plane:

Now let be one of the two critical values, so U1U1 U2U2 U4U4 U5U5 U0U0 U3U3 D And choose a disk D in one of the “symmetry sectors” in the parameter plane: Then G maps D univalently over all of U 0, so we again get a copy of the Cantor set in D G

Now let be one of the two critical values, so U1U1 U2U2 U4U4 U5U5 U0U0 U3U3 D And choose a disk D in one of the “symmetry sectors” in the parameter plane: Then G maps D univalently over all of U 0, so we again get a copy of the Cantor set in D G Then adjoining the appropriate Sierpinski holes gives a Cantor web in the parameter plane.

3. “Mandelpinski” necklaces A Mandlepinski necklace is a simple closed curve in the parameter plane that passes alternately through a certain number of centers of baby M-sets and the same number of centers of S-holes.

oops, sorry.... A Mandlepinski necklace is a simple closed curve in the parameter plane that passes alternately through a certain number of centers of baby M-sets and the same number of centers of S-holes. 3. “Mandelpinski” necklaces

A Mandlepinski necklace is a simple closed curve in the parameter plane that passes alternately through a certain number of centers of baby M-sets and the same number of centers of Sierpinski-holes. 3. “Mandelpinski” necklaces

A Julia setparameter plane n = 4 3. “Mandelpinski” necklaces

There is a “ring” around T passing through 8 = 2*4 preimages of T parameter plane n = 4 3. “Mandelpinski” necklaces

parameter plane n = 4 There is a “ring” around T passing through 8 = 2*4 preimages of T 3. “Mandelpinski” necklaces

Another “ring” around T passing through 32 = 2*4 2 preimages of T parameter plane n = 4 3. “Mandelpinski” necklaces

Another “ring” around T passing through 32 = 2*4 2 preimages of T parameter plane n = 4 3. “Mandelpinski” necklaces

Another “ring” around T passing through 128 = 2*4 3 preimages of T 3. “Mandelpinski” necklaces parameter plane n = 4

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* Sierpinski holes and 3 Mandelbrot sets

There is a ring around M that passes alternately through the centers of 3 = 2* Sierpinski holes and 3 Mandelbrot sets Now look around the McMullen domain in the parameter plane:

Another ring around M that passes alternately through the centers of 9 = 2* Sierpinski holes and 9 “Mandelbrot sets”* Now look around the McMullen domain in the parameter plane: *well, 3 period 2 bulbs

Another ring around M that passes alternately through the centers of 9 = 2* Sierpinski holes and 9 “Mandelbrot sets”* Now look around the McMullen domain in the parameter plane: *well, 3 period 2 bulbs

Then 33 = 2* Sierpinski holes and 33 Mandelbrot sets Now look around the McMullen domain in the parameter plane:

Then 129 = 2* Sierpinski holes and 129 Mandelbrot sets Now look around the McMullen domain in the parameter plane:

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: S 0 : 2 = 1* Sierpinski holes & M-sets S0S0 n = 3

S 1 : 4 = 1* Sierpinski holes & M-sets* *well, two period 2 bulbs n = 3

S 2 : 10 = 1* Sierpinski holes & “M-sets” n = 3

S 3 : 28 = 1* Sierpinski holes & M-sets n = 3

82, 244, then 730 Sierpinski holes... n = 3

the 13 th ring passes through 1,594,324 Sierpinski holes... n = 3 sorry, I forgot..... nevermind

* with one exception Theorem: For each n > 2, the McMullen domain is surrounded by infinitely many simple closed curves S k (“Mandelpinski” necklaces) having the property that: 1.each S k surrounds the McMullen domain and S k+1, and the S k accumulate on the boundary of M; 2.S k meets the center of exactly (n-2)n k Sierpinski holes, each with escape time k + 2; 3.S k also passes through the centers of the same number of baby Mandelbrot sets*

The critical points and prepoles all lie on the “critical circle” p c p c

The critical points and prepoles all lie on the “critical circle” The critical circle is mapped 2n-to-1 onto the “critical value ray” v 0 p c p c

The critical points and prepoles all lie on the “critical circle” The critical circle is mapped 2n-to-1 onto the “critical value ray” v 0 And every other circle centered at the origin and outside the critical circle is mapped n-to-1 to an ellipse with foci at the critical values

The critical points and prepoles all lie on the “critical circle” The critical circle is mapped 2n-to-1 onto the “critical value ray” v 0 And every other circle centered at the origin and outside the critical circle is mapped n-to-1 to an ellipse with foci at the critical values

The critical points and prepoles all lie on the “critical circle” The critical circle is mapped 2n-to-1 onto the “critical value ray” v 0 And every other circle centered at the origin and outside the critical circle is mapped n-to-1 to an ellipse with foci at the critical values, and same inside

There are no critical points outside the critical circle, so this region is mapped as n-to-1 covering onto the complement of the critical value ray. v 0

v 0 The interior of the critical circle is also mapped n-to-1 onto the complement of the critical value ray There are no critical points outside the critical circle, so this region is mapped as n-to-1 covering onto the complement of the critical value ray.

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

The dividing circle contains all parameters for which the critical values lie on the critical circle, i.e., When n = 4, the dividing circle passes through 3 centers of Sierpinski holes and 3 baby Mandelbrot sets

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

Reason: parameter plane n = 4 dynamical plane

Reason: as runs once around the dividing circle, parameter plane n = 4 dynamical plane

Reason: as runs once around the dividing circle, rotates 1/2 of a turn, parameter plane n = 4 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. parameter plane n = 4 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. So meets 3 prepoles and critical points enroute. parameter plane n = 4 dynamical plane

So the ring S 0 is just the dividing circle in parameter plane. S0S0 n = 4

So the ring S 0 is just the dividing circle in parameter plane. S0S0 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

so maps the critical circle C 0 strictly inside itself C0C0

Now there is a preimage C 1 of the critical circle that is mapped 4-to-1 onto the critical circle, and this curve contains 32 pre-critical points and 32 pre-pre-poles. C0C0 C1C1

And then a preimage C 2 of the C 1 that is mapped 4-to-1 onto the C 1, and so 16-to-1 onto C 0, and this curve contains 128 pre-pre-critical points and 128 pre-pre-pre-poles, etc. C0C0 C1C1 C2C2

The rings C 0 and C 1

Let be the second iterate of the critical point

Let be the second iterate of the critical point So when n = 4.

Let be the second iterate of the critical point So when n = 4. Note that as provided n > 2. When n = 2,, a very different situation.

G maps points in the parameter plane to points in the dynamical plane C0C0 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 C 0 G(D) C0C0 D

and G(D) covers C 1, C 2,... C0C0 D G

Choose a small disk D 0 inside M. Then G maps the annulus A = D - D 0 univalently over all of the C j, j > 0. C0C0 D0D0 A G

Choose a parametrization of C k, say. So we have a second map from A into G(A), C0C0 D0D0 A G

Since G is 1-to-1, we thus have a map which takes A into A. C0C0 D0D0 AH

C0C0 D0D0 AH Let S be the covering strip of A and let H*: S S be the covering map of H: A A

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.

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. So the map gives a parametrization of the ring S k in the parameter plane, and -values that correspond to pre-poles or pre-critical points are then parameters at the centers of Sierpinski holes or baby Mandelbrot sets.

There are (n - 2)n k-1 pre-poles in the k th dynamical plane ring, but (n - 2)n k centers of Sierpinski holes in the parameter plane rings. Here’s the reason:

On the annulus A, There are (n - 2)n k-1 pre-poles in the k th dynamical plane ring, but (n - 2)n k centers of Sierpinski holes in the parameter plane rings. Here’s the reason:

On the annulus A, So as rotates clockwise around the ring S k, rotates once around the origin in the counterclockwise direction. Meanwhile, each pre-pole and pre-critical point rotates clockwise by approximately 1/((n-2)n k-1 of a turn. There are (n - 2)n k-1 pre-poles in the k th dynamical plane ring, but (n - 2)n k centers of Sierpinski holes in the parameter plane rings. Here’s the reason:

On the annulus A, So hits one additional prepole or pre-critical point while traveling around each S k. There are (n - 2)n k-1 pre-poles in the k th dynamical plane ring, but (n - 2)n k centers of Sierpinski holes in the parameter plane rings. Here’s the reason: So as rotates clockwise around the ring S k, rotates once around the origin in the counterclockwise direction. Meanwhile, each pre-pole and pre-critical point rotates clockwise by approximately 1/((n-2)n k-1 of a turn.

Similar arguments show that each Sierpinski hole on a Mandelpinski necklace is also surrounded by infinitely many sub-Mandelpinski necklaces

Similar arguments show that each Sierpinski hole on a Mandelpinski necklace is also surrounded by infinitely many sub-Mandelpinski necklaces

Similar arguments show that each Sierpinski hole on a Mandelpinski necklace is also surrounded by infinitely many sub-Mandelpinski necklaces

Similar arguments show that each Sierpinski hole on a Mandelpinski necklace is also surrounded by infinitely many sub-Mandelpinski necklaces

Similar arguments show that each Sierpinski hole on a Mandelpinski necklace is also surrounded by infinitely many sub-Mandelpinski necklaces

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

Some open problems: 1.By Xiaoguang Wang’s result yesterday, the boundary of B is always a simple closed curve (except when J is a Cantor set) when n > 2. Is the boundary of the parameter plane also a simple closed curve??? n = 4

Some open problems: 1.By Xiaoguang Wang’s result yesterday, the boundary of B is always a simple closed curve (except when J is a Cantor set) when n > 2. Is the boundary of the parameter plane also a simple closed curve??? 2.What about the crazy case n = 2???

Some open problems: 1.By Xiaoguang Wang’s result yesterday, the boundary of B is always a simple closed curve (except when J is a Cantor set) when n > 2. Is the boundary of the parameter plane also a simple closed curve??? 2.What about the crazy case n = 2??? 3.Are the Julia sets for these maps always locally connected?

Some open problems: 1.By Xiaoguang Wang’s result yesterday, the boundary of B is always a simple closed curve (except when J is a Cantor set) when n > 2. Is the boundary of the parameter plane also a simple closed curve??? 2.What about the crazy case n = 2??? 3.Are the Julia sets for these maps always locally connected? 4. Are the parameter planes locally connected???

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

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

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 n = 2, d = 1 Not a baby M-set

n = 2, d = 1 No Cantor necklace 7. What is going on in the parameter plane for the maps Not a baby M-set

n = 2, d = 1 No Cantor necklace 7. What is going on in the parameter plane for the maps

n = 4, d = 1 7. What is going on in the parameter plane for the maps J approaches the unit disk only along these 3 lines