Singular Perturbations of Complex Polynomials Dynamics of the family of complex maps with: Paul Blanchard Toni Garijo Matt Holzer Dan Look Sebastian Marotta Mark Morabito Monica Moreno Rocha Kevin Pilgrim Elizabeth Russell Yakov Shapiro David Uminsky Sum Wun Ellce

Singular Perturbations of Complex Polynomials 1. The Escape Trichotomy Cantor set Sierpinski curve Cantor set of circles

Singular Perturbations of Complex Polynomials 2. Structures in the dynamical plane All are Sierpinski curve Julia sets

Singular Perturbations of Complex Polynomials 3. Structures in the parameter planes

These lectures will deal with the dynamics of the family of complex maps where c is the center of a hyperbolic component of the Multibrot set

These lectures will deal with the dynamics of the family of complex maps where c is the center of a hyperbolic component of the Multibrot set But for simplicity, we’ll concentrate for the most part on the easier family

Why the interest in these maps?

Why the interest in these maps? First, these are singular perturbations of .

Why the interest in these maps? First, these are singular perturbations of . When but when , the dynamical behavior “explodes.” we completely understand the dynamics of

Why the interest in these maps? First, these are singular perturbations of . Second, how do you solve the equation ?

Why the interest in these maps? First, these are singular perturbations of . Second, how do you solve the equation ? You use Newton’s method (of course!):

Why the interest in these maps? First, these are singular perturbations of . Second, how do you solve the equation ? You use Newton’s method (of course!): Iterate:

Why the interest in these maps? First, these are singular perturbations of . Second, how do you solve the equation ? You use Newton’s method (of course!): Iterate: a singular perturbation of z/2

Why the interest in these maps? First, these are singular perturbations of . Second, how do you solve the equation ? You use Newton’s method (of course!): Whenever the equation has a multiple root, the corresponding Newton’s method involves a singular perturbation.

Why the interest in these maps? First, these are singular perturbations of . Second, how do you solve the equation ? Third, we are looking at maps on the boundary of the set of rational maps of degree 2n --- a very interesting topic of contemporary research.

Dynamics of the family of complex maps The Escape Trichotomy Dynamics of the family of complex maps with: Paul Blanchard Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta Monica Moreno Rocha Yakov Shapiro David Uminsky

A rational map of degree 2n. Dynamics of complex and A rational map of degree 2n.

Dynamics of complex and A rational map of degree 2n. The Julia set is: The closure of the set of repelling periodic points; The boundary of the escaping orbits; The chaotic set. The Fatou set is the complement of .

When , the Julia set is the unit circle

When , the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When , the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When , the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When , the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When , the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When , the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When , the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When , the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When , the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When , the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When , the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When , the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When , the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When , the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When , the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When , the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When , the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When , the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When , the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When , the Julia set is the unit circle But when , the Julia set explodes

When , the Julia set is the unit circle But when , the Julia set explodes A Sierpinski curve

When , the Julia set is the unit circle But when , the Julia set explodes Another Sierpinski curve

When , the Julia set is the unit circle But when , the Julia set explodes Also a Sierpinski curve

The Julia set has 2n-fold symmetry since Easy computations: The Julia set has 2n-fold symmetry since where is a 2nth root of unity

Easy computations: 2n free critical points

Easy computations: 2n free critical points

Easy computations: 2n free critical points Only 2 critical values

Easy computations: 2n free critical points Only 2 critical values

Easy computations: 2n free critical points Only 2 critical values

Easy computations: 2n free critical points Only 2 critical values But really only 1 free critical orbit since the map has 2n-fold symmetry

Easy computations: B is superattracting, so have immediate basin B mapped n-to-1 to itself.

Easy computations: B is superattracting, so have immediate basin B mapped n-to-1 to itself. T 0 is a pole, so have trap door T mapped n-to-1 to B.

So any orbit that eventually Easy computations: B is superattracting, so have immediate basin B mapped n-to-1 to itself. T 0 is a pole, so have trap door T mapped n-to-1 to B. So any orbit that eventually enters B must do so by passing through T.

There are three distinct ways the critical orbit can enter B: The Escape Trichotomy There are three distinct ways the critical orbit can enter B:

There are three distinct ways the critical orbit can enter B: The Escape Trichotomy There are three distinct ways the critical orbit can enter B: B is a Cantor set

The Escape Trichotomy There are three distinct ways the critical orbit can enter B: B is a Cantor set T is a Cantor set of simple closed curves (this case does not occur if n = 2)

The Escape Trichotomy There are three distinct ways the critical orbit can enter B: B is a Cantor set T is a Cantor set of simple closed curves (this case does not occur if n = 2) T is a Sierpinski curve

parameter plane when n = 3 Case 1: B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

parameter plane when n = 3 J is a Cantor set B is a Cantor set

B is a Cantor set Draw curves connecting the two critical values to in B

B is a Cantor set The preimages are curves passing through the critical points and connecting c to

B is a Cantor set The preimages are curves passing through the critical points and connecting c to and to 0

Choose a large circle in B is a Cantor set Choose a large circle in B

and locate its two preimages B is a Cantor set and locate its two preimages

Construct the regions I0, ... I2n-1 B is a Cantor set I1 I2 I0 I3 I5 I4 Construct the regions I0, ... I2n-1

B is a Cantor set I1 I2 I0 I3 I5 I4 Each of the regions I0, ... I2n-1 is mapped 1-1 over the union of the Ij

B is a Cantor set I1 I2 I0 I3 I5 I4 Each of the regions I0, ... I2n-1 is mapped 1-1 over the union of the Ij

B is a Cantor set I1 I2 I0 I3 I5 I4 Each of the regions I0, ... I2n-1 is mapped 1-1 over the union of the Ij

B is a Cantor set I1 I2 I0 I3 I5 I4 Each of the regions I0, ... I2n-1 is mapped 1-1 over the union of the Ij

B is a Cantor set I1 I2 I0 I3 I5 I4 Each of the regions I0, ... I2n-1 is mapped 1-1 over the union of the Ij is a Cantor set

Case 2: the critical values lie in T, not B parameter plane when n = 3

lies in the McMullen domain parameter plane when n = 3 T lies in the McMullen domain

lies in the McMullen domain parameter plane when n = 3 J is a Cantor set of simple closed curves T lies in the McMullen domain Remark: There is no McMullen domain in the case n = 2.

lies in the McMullen domain parameter plane when n = 3 J is a Cantor set of simple closed curves T lies in the McMullen domain

lies in the McMullen domain parameter plane when n = 3 J is a Cantor set of simple closed curves T lies in the McMullen domain

lies in the McMullen domain parameter plane when n = 3 J is a Cantor set of simple closed curves T lies in the McMullen domain

lies in the McMullen domain parameter plane when n = 3 J is a Cantor set of simple closed curves T lies in the McMullen domain

lies in the McMullen domain parameter plane when n = 3 J is a Cantor set of simple closed curves T lies in the McMullen domain

lies in the McMullen domain parameter plane when n = 3 J is a Cantor set of simple closed curves T lies in the McMullen domain

J is a Cantor set of “circles” v c Why is the preimage of T an annulus?

Could it be that each critical point lies in a disjoint preimage of T?

Could it be that each critical point lies in a disjoint preimage of T? No. The map would then be 4n to 1 on the preimage of T.

By 2n-fold symmetry, there can then be only Could it be that each critical point lies in a disjoint preimage of T? No. The map would then be 4n to 1 on the preimage of T. By 2n-fold symmetry, there can then be only one preimage of T

Riemann-Hurwitz formula: F: D R holomorphic branched covering n(D), n(R) = number of boundary components of D, R

Riemann-Hurwitz formula: F: D R holomorphic branched covering n(D), n(R) = number of boundary components of D, R then: n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) *

In our case, R is the trap door and D is the Riemann-Hurwitz formula: F: D R holomorphic branched covering n(D), n(R) = number of boundary components of D, R then: n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) * In our case, R is the trap door and D is the preimage of T

Riemann-Hurwitz formula: F: D R holomorphic branched covering n(D), n(R) = number of boundary components of D, R then: n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) * We have deg F = 2n on D, so n(D) - 2 = ( 2n ) (n(R) - 2) + (# of critical points in D) *

and R = T is a disk, so n(R) = 1 Riemann-Hurwitz formula: F: D R holomorphic branched covering n(D), n(R) = number of boundary components of D, R then: n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) * and R = T is a disk, so n(R) = 1 n(D) - 2 = ( 2n ) (1 - 2) + (# of critical points in D) *

and there are 2n critical points in D Riemann-Hurwitz formula: F: D R holomorphic branched covering n(D), n(R) = number of boundary components of D, R then: n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) * and there are 2n critical points in D n(D) - 2 = ( 2n ) (1 - 2) + ( 2n ) = 0 *

and there are 2n critical points in D Riemann-Hurwitz formula: F: D R holomorphic branched covering n(D), n(R) = number of boundary components of D, R then: n(D) - 2 = (deg F) (n(R) - 2) + (# of critical points in D) * and there are 2n critical points in D n(D) - 2 = ( 2n ) (1 - 2) + ( 2n ) = 0 * so n(D) = 2 and the preimage of T is an annulus.

J is a Cantor set of “circles” v c So the preimage of T is an annulus.

J is a Cantor set of “circles” B T B and T are mapped n-to-1 to B

J is a Cantor set of “circles” B T The white annulus is mapped 2n-to-1 to T

J is a Cantor set of “circles” B T So all that’s left are the blue annuli, and each are mapped n-to-1 to the union of the blue and white annuli.

J is a Cantor set of “circles” B T So there are sub-annuli in the blue annuli that are mapped onto the white annulus.

J is a Cantor set of “circles”

J is a Cantor set of “circles”

J is a Cantor set of “circles” then all other preimages of F-1(T) contain no critical points, and F is an n - to -1 covering on each, so the remaining preimages of T are all annuli.

J is a Cantor set of “circles” These annuli fill out the Fatou set; removing all of them leaves us with a Cantor set of simple closed curves (McMullen)

Curiously, this cannot happen when n = 2.

Reason: Modulus of an annulus 1 r A Any annulus in the plane is conformally isomorphic to a unique round annulus of the form .

Reason: Modulus of an annulus 1 r A Any annulus in the plane is conformally isomorphic to a unique round annulus of the form . Then the modulus of A is

Reason: Modulus of an annulus 1 r A Any annulus in the plane is conformally isomorphic to a unique round annulus of the form . Then the modulus of A is And if F : A1 A2 is an analytic n-to-1 covering map, then mod (A2) = n mod (A1)

This cannot happen when n = 2 Aout Amid Ain 2-to-1,

This cannot happen when n = 2 Aout Amid Ain 2-to-1, so the modulus of is 1/2 the modulus of

This cannot happen when n = 2 Aout Amid Ain 2-to-1, so the modulus of is 1/2 the modulus of , same with

This cannot happen when n = 2 Aout Amid Ain But then mod ( ) + mod ( ) = mod ( ) so there is no room for the middle annulus

Another reason this does not happen: The critical values are When n > 2 we have

Another reason this does not happen: The critical values are When n > 2 we have so the critical value lies in T when is small.

Another reason this does not happen: The critical values are But when n = 2 we have

Another reason this does not happen: The critical values are But when n = 2 we have So, as , and 1/4 is nowhere near the basin of when is small

Parameter planes n = 2 n = 3 No McMullen domain McMullen domain

Parameter planes n = 2 n = 3 No McMullen domain McMullen domain

n = 2 n = 3 No McMullen domain McMullen domain

n = 2 n = 3 No McMullen domain McMullen domain

There is a lot of structure around the McMullen domain when n > 2 but a very different structure when n = 2.

Case 3: the critical orbit eventually lands in the trap door is a “Sierpinski curve.”

Sierpinski Curve A Sierpinski curve is any planar set that is homeomorphic to the Sierpinski carpet fractal. The Sierpinski Carpet

Topological Characterization Any planar set that is: 1. compact 2. connected 3. locally connected 4. nowhere dense 5. any two complementary domains are bounded by simple closed curves that are pairwise disjoint is a Sierpinski curve. The Sierpinski Carpet

A Sierpinski curve is a universal plane continuum: More importantly.... A Sierpinski curve is a universal plane continuum: Any planar, one-dimensional, compact, connected set can be homeomorphically embedded in a Sierpinski curve. For example....

The topologist’s sine curve can be embedded inside

The topologist’s sine curve can be embedded inside

The topologist’s sine curve can be embedded inside skip Knaster

The Knaster continuum can be embedded inside

Start with the Cantor middle-thirds set Knaster continuum A well known example of an indecomposable continuum Start with the Cantor middle-thirds set

Knaster continuum Connect symmetric points about 1/2 with semicircles

Knaster continuum Do the same below about 5/6

Knaster continuum And continue....

Knaster continuum

Properties of K: There is one curve that passes through all the endpoints of the Cantor set.

Properties of K: There is one curve that passes through all the endpoints of the Cantor set. It accumulates everywhere on itself and on K.

Properties of K: There is one curve that passes through all the endpoints of the Cantor set. It accumulates everywhere on itself and on K. And is the only piece of K that is accessible from the outside.

Properties of K: There is one curve that passes through all the endpoints of the Cantor set. It accumulates everywhere on itself and on K. And is the only piece of K that is accessible from the outside. But there are infinitely many other curves in K, each of which is dense in K.

The Knaster continuum can be embedded inside

Note the Cantor middle-thirds set along the middle horizontal line The Knaster continuum can be embedded inside Note the Cantor middle-thirds set along the middle horizontal line

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

Even this “curve” can be embedded inside

can be embedded inside Even this “curve” Moreover, Sierpinski curves occur all the time as Julia sets.

lies in a Sierpinski hole Case 3: the critical orbit eventually lands in the trap door. parameter plane when n = 3 T lies in a Sierpinski hole

lies in a Sierpinski hole parameter plane when n = 3 J is a Sierpinski curve T lies in a Sierpinski hole

lies in a Sierpinski hole parameter plane when n = 3 J is a Sierpinski curve T lies in a Sierpinski hole

lies in a Sierpinski hole parameter plane when n = 3 J is a Sierpinski curve T lies in a Sierpinski hole

lies in a Sierpinski hole parameter plane when n = 3 J is a Sierpinski curve T lies in a Sierpinski hole

lies in a Sierpinski hole parameter plane when n = 3 J is a Sierpinski curve T lies in a Sierpinski hole

lies in a Sierpinski hole parameter plane when n = 3 J is a Sierpinski curve T lies in a Sierpinski hole

lies in a Sierpinski hole parameter plane when n = 3 J is a Sierpinski curve T lies in a Sierpinski hole

lies in a Sierpinski hole parameter plane when n = 3 J is a Sierpinski curve T lies in a Sierpinski hole

lies in a Sierpinski hole parameter plane when n = 3 J is a Sierpinski curve T lies in a Sierpinski hole

lies in a Sierpinski hole parameter plane when n = 3 J is a Sierpinski curve T lies in a Sierpinski hole

lies in a Sierpinski hole parameter plane when n = 3 J is a Sierpinski curve T lies in a Sierpinski hole

To show that is homeomorphic to

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s Fatou set is the union of the preimages of B; all disjoint, open disks.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s Fatou set is the union of the preimages of B; all disjoint, open disks.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s If J contains an open set, then J = C.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s If J contains an open set, then J = C.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s No recurrent critical orbits, no parabolic points., and F is hyperbolic on J....

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s No recurrent critical orbits, no parabolic points., and F is hyperbolic on J....

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s J locally connected, so the boundaries are locally connected. Need to show they are s.c.c.’s. Can only meet at (preimages of) critical points, hence disjoint.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s So J is a Sierpinski curve.

Have an exact count of the number of Sierpinski holes: Theorem (Roesch): Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3)

Have an exact count of the number of Sierpinski holes: Theorem (Roesch): Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3) Reason: The equation reduces to a polynomial of degree (n-1)(2n)(k-3) , and it can be shown that all the roots of this polynomial are distinct. (You can put a Böttcher coordinate on each Sierpinski hole).

Have an exact count of the number of Sierpinski holes: Theorem (Roesch): Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3) Reason: The equation reduces to a polynomial of degree (n-1)(2n)(k-3) , and it can be shown that all the roots of this polynomial are distinct. (You can put a Böttcher coordinate on each Sierpinski hole). So we have exactly that many “centers” of Sierpinski holes, i.e., parameters for which the critical points all land on 0 and then .

Have an exact count of the number of Sierpinski holes: Theorem (Roesch): Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3) n = 3 escape time 3 2 Sierpinski holes parameter plane n = 3

Have an exact count of the number of Sierpinski holes: Theorem (Roesch): Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3) n = 3 escape time 3 2 Sierpinski holes parameter plane n = 3

Have an exact count of the number of Sierpinski holes: Theorem (Roesch): Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3) n = 3 escape time 4 12 Sierpinski holes parameter plane n = 3

Have an exact count of the number of Sierpinski holes: Theorem (Roesch): Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3) n = 3 escape time 4 12 Sierpinski holes parameter plane n = 3

Have an exact count of the number of Sierpinski holes: Theorem (Roesch): Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3) n = 4 escape time 3 3 Sierpinski holes parameter plane n = 4

Have an exact count of the number of Sierpinski holes: Theorem (Roesch): Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3) n = 4 escape time 4 24 Sierpinski holes parameter plane n = 4

Have an exact count of the number of Sierpinski holes: Theorem (Roesch): Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3) n = 4 escape time 12 402,653,184 Sierpinski holes Sorry. I forgot to indicate their locations. parameter plane n = 4

Given two Sierpinski curve Julia sets, when do we know that the dynamics on them are the same, i.e., the maps are conjugate on the Julia sets? Main Question: These sets are homeomorphic, but are the dynamics on them the same?

Given two Sierpinski curve Julia sets, when do we know that the dynamics on them are the same, i.e., the maps are conjugate on the Julia sets? Main Question: We know when two maps with Sierpinski curve Julia sets are conjugate on their Julia sets (next lecture), but: Problem: What is the different dynamical behavior when two such maps are not conjugate on their Julia sets. Clearly a job for symbolic dynamics.....

Other ways that Sierpinski curve Julia sets arise: Buried points in Cantor necklaces; 2. Main cardioids in buried Mandelbrot sets; Cantor sets of circles around the McMullen domain; 4. Other families of rational maps; 5. Sierpinski gaskets 6. Major applications

1. Cantor necklaces in the parameter plane The necklace is the Cantor middle thirds set with disks replacing removed intervals. parameter plane n = 4

1. Cantor necklaces in the parameter plane There is a Cantor necklace along the negative real axis parameter plane n = 4

1. Cantor necklaces in the parameter plane 5 5 4 3 4 The open disks are Sierpinski holes parameter plane n = 4

1. Cantor necklaces in the parameter plane 5 5 4 3 4 The open disks are Sierpinski holes; the buried points in the Cantor set also correspond to Sierpinski curves; parameter plane n = 4

1. Cantor necklaces in the parameter plane 5 5 4 3 4 The open disks are Sierpinski holes; the buried points in the Cantor set also correspond to Sierpinski curves; and all are dynamically different. parameter plane n = 4

1. Cantor necklaces in the parameter plane A similar Cantor necklace when n = 2 The parameter plane when n = 2

1. Cantor necklaces in the parameter plane A similar Cantor necklace when n = 2 The parameter plane when n = 2

1. Cantor necklaces in the parameter plane 6 6 5 4 5 A similar Cantor necklace when n = 2 The parameter plane when n = 2

1. Cantor necklaces in the parameter plane When the parameter is negative, the critical points do not lie on the real axis; A Sierpinski curve from a buried point in the Cantor necklace when n=2

1. Cantor necklaces in the parameter plane When the parameter is negative, the critical points do not lie on the real axis; A Sierpinski curve from a buried point in the Cantor necklace when n=2

1. Cantor necklaces in the parameter plane When the parameter is negative, the critical points do not lie on the real axis; they map to the imaginary axis; A Sierpinski curve from a buried point in the Cantor necklace when n=2

1. Cantor necklaces in the parameter plane When the parameter is negative, the critical points do not lie on the real axis; they map to the imaginary axis; and then to the real axis, where the orbit then remains forever. A Sierpinski curve from a buried point in the Cantor necklace when n=2

1. Cantor necklaces in the parameter plane When the parameter is negative, the critical points do not lie on the real axis; they map to the imaginary axis; and then to the real axis, where the orbit then remains forever. So the critical orbits are non-recurrrent J is locally connected. A Sierpinski curve from a buried point in the Cantor necklace when n=2

1. Cantor necklaces in the parameter plane The only Fatou components are B and its preimages since the critical orbits lie in a Cantor set which is not in the Fatou set J is compact and connected. A Sierpinski curve from a buried point in the Cantor necklace when n=2

1. Cantor necklaces in the parameter plane There is a Fatou component J is nowhere dense A Sierpinski curve from a buried point in the Cantor necklace when n=2

1. Cantor necklaces in the parameter plane There is a Fatou component J is nowhere dense, and same arguments as before show that the boundary consists of disjoint simple closed curves J is a Sierpinski curve. A Sierpinski curve from a buried point in the Cantor necklace when n=2

1. Cantor necklaces in the parameter plane The “endpoints” in the Cantor set (parameters on the boundaries of the Sierpinski holes) do not correspond to Sierpinski curves. A “hybrid” Sierpinski curve; some boundary curves meet.

1. Cantor necklaces in the parameter plane The “endpoints” in the Cantor set (parameters on the boundaries of the Sierpinski holes) do not correspond to Sierpinski curves. Certain preimages of T touch each other at (pre)-critical points. A “hybrid” Sierpinski curve; some boundary curves meet.

1. Cantor necklaces in the parameter plane There are lots of other Cantor necklaces in the parameter planes. parameter plane n = 2

1. Cantor necklaces in the parameter plane There are lots of other Cantor necklaces in the parameter planes. parameter plane n = 2

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve. A Sierpinski curve, but very different dynamically from the earlier ones. n = 4

If n > 2, there are uncountably many simple closed curves surrounding the McMullen domain; all these parameters are Sierpinski curves. n = 3

If n > 2, there are uncountably many simple closed curves surrounding the McMullen domain; all these parameters are Sierpinski curves. n = 3

If n > 2, there are uncountably many simple closed curves surrounding the McMullen domain; all these parameters are Sierpinski curves. All parameters on these curves correspond to Sierpinski curves. n = 3

If n > 2, there are uncountably many simple closed curves surrounding the McMullen domain; all these parameters are Sierpinski curves. And all non-symmetrically located parameters on these curves have very different dynamics on J. n = 3

If n > 2, there are also countably many different simple closed curves accumulating on the McMullen domain. Each alternately passes through centers of Sierpinski holes and centers of baby Mandelbrot sets. 2 centers n = 3

If n > 2, there are also countably many different simple closed curves accumulating on the McMullen domain. Each alternately passes through centers of Sierpinski holes and centers of baby Mandelbrot sets. 4 centers n = 3

If n > 2, there are also countably many different simple closed curves accumulating on the McMullen domain. Each alternately passes through centers of Sierpinski holes and centers of baby Mandelbrot sets. 10 centers n = 3

If n > 2, there are also countably many different simple closed curves accumulating on the McMullen domain. Each alternately passes through centers of Sierpinski holes and centers of baby Mandelbrot sets. 28 centers n = 3

If n > 2, there are also countably many different simple closed curves accumulating on the McMullen domain. Each alternately passes through centers of Sierpinski holes and centers of baby Mandelbrot sets. 82 centers n = 3

If n > 2, there are also countably many different simple closed curves accumulating on the McMullen domain. Each alternately passes through centers of Sierpinski holes and centers of baby Mandelbrot sets. passes through 1,594,324 centers. n = 3

If n > 2, there are also countably many different simple closed curves accumulating on the McMullen domain. Each alternately passes through centers of Sierpinski holes and centers of baby Mandelbrot sets. As before, all non- symmetrically located centers have different dynamics. n = 3

Consider the family of maps where c is the center of a hyperbolic component of the Mandelbrot set.

Consider the family of maps where c is the center of a hyperbolic component of the Mandelbrot set.

Consider the family of maps where c is the center of a hyperbolic component of the Mandelbrot set.

, the Julia set again expodes. When , the Julia set again expodes.

, the Julia set again expodes. When , the Julia set again expodes.

, the Julia set again expodes. When , the Julia set again expodes.

, the Julia set again expodes. When , the Julia set again expodes.

, the Julia set again expodes. When , the Julia set again expodes. A doubly-inverted Douady rabbit.

If you chop off the “ears” of each internal rabbit in each component of the original Julia set, then what’s left is another Sierpinski curve (provided that both of the critical orbits eventually escape).

The (generalized) Sierpinski gasket also occurs (although rarely) as a Julia set in these families

Choose at the tip of the parameter plane on the negative real axis Parameter plane for n=2, d=1

This Julia set is homeomorphic to the gasket

If is a Misiurewicz parameter on the boundary of the connectedness locus, then is a “generalized” gasket. Parameter plane for n=2, d=1

If is a Misiurewicz parameter on the boundary of the connectedness locus, then is a “generalized” gasket. Parameter plane for n=2, d=1

If is a Misiurewicz parameter on the boundary of the connectedness locus, then is a “generalized” gasket. Parameter plane for n=2, d=1

None of these other Julia sets are homeomorphic to the Sierpinski gasket, however.

None of these other Julia sets are homeomorphic to the Sierpinski gasket, however.

None of these other Julia sets are homeomorphic to the Sierpinski gasket, however. T The “triangle” T touches the inner triangle twice, not once.

For other values of n and d, the generalized Sierpinski gaskets are not triangular. T n = d = 2 A “square” gasket

For other values of n and d, the generalized Sierpinski gaskets are not triangular. n = 3, d = 2 A “pentagonal” gasket

Here’s the parameter plane when n = 2:

and this object appears everywhere..... Rotate it by 90 degrees: and this object appears everywhere.....