Dynamic Classification of Escape Time Sierpinski Curve Julia Sets Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer U. Hoomiforgot Dan Look Sebastian Marotta with: Mark Morabito Monica Moreno Rocha Kevin Pilgrim Elizabeth Russell Yakov Shapiro David Uminsky, n > 1

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

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.

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

This set

can be embedded inside

This set can be embedded inside Moreover, Sierpinski curves occur all the time as Julia sets.

Dynamics of complex and A rational map of degree 2n. Also a “singular perturbation” of z n.

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 When, the Julia set is the unit circle

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 is symmetric under

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

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

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

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

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

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

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

B is a Cantor set 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 and no parabolic points.

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

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: Reason: The equation reduces to a polynomial of degree (n-1)(2n) (k-3) ; we have a Bottcher coordinate on each Sierpinski hole; and so all the roots of this polynomial are distinct. So we have exactly that many “centers” of Sierpinski holes, i.e., parameters for which the critical points all land on 0 and then on. 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: n = 3 escape time 3 2 Sierpinski holes parameter plane n = 3 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: n = 3 escape time 3 2 Sierpinski holes parameter plane n = 3 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: n = 3 escape time 4 12 Sierpinski holes parameter plane n = 3 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: n = 3 escape time 4 12 Sierpinski holes parameter plane n = 3 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: n = 4 escape time 3 3 Sierpinski holes parameter plane n = 4 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: n = 4 escape time 4 24 Sierpinski holes parameter plane n = 4 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: n = 4 escape time ,653,184 Sierpinski holes Sorry. I forgot to indicate their locations. parameter plane n = 4 Theorem (Roesch): Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3)

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?

#1: If and are drawn from the same Sierpinski hole, then the corresponding maps have the same dynamics, i.e., they are topologically conjugate on their Julia sets. parameter plane n = 4

#1: If and are drawn from the same Sierpinski hole, then the corresponding maps have the same dynamics, i.e., they are topologically conjugate on their Julia sets. So all these parameters have the same dynamics on their Julia sets. parameter plane n = 4

#1: If and are drawn from the same Sierpinski hole, then the corresponding maps have the same dynamics, i.e., they are topologically conjugate on their Julia sets. parameter plane n = 4 This uses quasiconformal surgery techniques

#1: If and are drawn from the same Sierpinski hole, then the corresponding maps have the same dynamics, i.e., they are topologically conjugate on their Julia sets. #2: If these parameters come from Sierpinski holes with different “escape times,” then the maps cannot be conjugate.

Two Sierpinski curve Julia sets, so they are homeomorphic.

escape time 3escape time 4 So these maps cannot be topologically conjugate.

is the only invariant boundary of an escape component, so must be preserved by any conjugacy.

is the only preimage of, so this curve must also be preserved by a conjugacy.

If a boundary component is mapped to after k iterations, its image under the conjugacy must also have this property, and so forth.....

The curves around c are special; they are the only other ones in J mapped 2-1 onto their images. c

This bounding region takes 3 iterates to land on the boundary of B. But this bounding region takes 4 iterates to land, so these maps are not conjugate.

For this it suffices to consider the centers of the Sierpinski holes; i.e., parameter values for which for some k  3. #3: What if two maps lie in different Sierpinski holes that have the same escape time?

#3: What if two maps lie in different Sierpinski holes that have the same escape time? For this it suffices to consider the centers of the Sierpinski holes; i.e., parameter values for which for some k  3. Two such centers of Sierpinski holes are “critically finite” maps, so by Thurston’s Theorem, if they are topologically conjugate in the plane, they can be conjugated by a Mobius transformation (in the orientation preserving case).

#3: What if two maps lie in different Sierpinski holes that have the same escape time? For this it suffices to consider the centers of the Sierpinski holes; i.e., parameter values for which for some k  3. Two such centers of Sierpinski holes are “critically finite” maps, so by Thurston’s Theorem, if they are topologically conjugate in the plane, they can be conjugated by a Mobius transformation (in the orientation preserving case). Since and under the conjugacy, the Mobius conjugacy must be of the form.

then: If we have a conjugacy

then: Comparing coefficients:

then: Comparing coefficients: If we have a conjugacy

then: Comparing coefficients: Easy check --- for the orientation reversing case: is conjugate to via If we have a conjugacy

Theorem. If and are centers of Sierpinski holes, then iff or where is a primitive root of unity; then any two parameters drawn from these holes have the same dynamics. n = 3:Only and are conjugate centers since,,,,,n = 4:Only are conjugate centers where.

n = 3, escape time 4, 12 Sierpinski holes, but only six conjugacy classes conjugate centers:,

n = 3, escape time 4, 12 Sierpinski holes, but only six conjugacy classes conjugate centers:,

n = 3, escape time 4, 12 Sierpinski holes, but only six conjugacy classes conjugate centers:,

n = 3, escape time 4, 12 Sierpinski holes, but only six conjugacy classes conjugate centers:,

,,,,,where n = 4, escape time 4, 24 Sierpinski holes, but only five conjugacy classes conjugate centers:

,,,,,where n = 4, escape time 4, 24 Sierpinski holes, but only five conjugacy classes conjugate centers:

,,,,,where n = 4, escape time 4, 24 Sierpinski holes, but only five conjugacy classes conjugate centers:

,,,,,where n = 4, escape time 4, 24 Sierpinski holes, but only five conjugacy classes conjugate centers:

,,,,,where n = 4, escape time 4, 24 Sierpinski holes, but only five conjugacy classes conjugate centers:

Theorem: For any n there are exactly (n-1) (2n) Sierpinski holes with escape time k. The number of distinct conjugacy classes is given by: k-3 a. (2n) when n is odd; k-3 b. (2n) /2 + 2 when n is even. k-3k-4

For n odd, there are no Sierpinski holes along the real axis, so there are exactly n - 1 conjugate Sierpinski holes. n = 3n = 5

For n even, there is a “Cantor necklace” along the negative axis, so there are some “real” Sierpinski holes, n = 4

For n even, there is a “Cantor necklace” along the negative axis, so there are some “real” Sierpinski holes, n = 4

For n even, there is a “Cantor necklace” along the negative axis, so there are some “real” Sierpinski holes, n = 4magnification M

For n even, there is a “Cantor necklace” along the negative axis, so there are some “real” Sierpinski holes, n = 4magnification M

For n even, there is a “Cantor necklace” along the negative axis, so we can count the number of “real” Sierpinski holes, and there are exactly n - 1 conjugate holes in this case: n = 4magnification M

For n even, there are also 2(n - 1) “complex” Sierpinski holes that have conjugate dynamics: n = 4magnification M

n = 4: 402,653,184 Sierpinski holes with escape time 12; 67,108,832 distinct conjugacy classes. Sorry. I again forgot to indicate their locations.

n = 4: 402,653,184 Sierpinski holes with escape time 12; 67,108,832 distinct conjugacy classes. Problem: Describe the dynamics on these different conjugacy classes.

Other ways that Sierpinski curve Julia sets arise: 1.Buried points in Cantor necklaces;Buried points in Cantor necklaces; 2. Main cardioids in buried Mandelbrot sets; 3.Structure around the McMullen domain;Structure around the McMullen domain; 4. Other families of rational maps; 5. The difference between n = 2 and n >2 ; 6. Major applicationsMajor applications

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

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

parameter plane n = 4 The open disks are Sierpinski holes Cantor necklaces in the parameter plane

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

parameter plane n = 4 The open disks are Sierpinski holes; the buried points in the Cantor set also correspond to Sierpinski curves; and all are dynamically different 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. 1. Cantor necklaces in the parameter plane A “hybrid” Sierpinski curve; some boundary curves meet.

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

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

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

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

n = 4 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

n = 4 A Sierpinski curve, but very different dynamically from the earlier ones. 2.If lies in the main cardioid of a buried Mandelbrot set, then again the Julia set is a Sierpinski curve.

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

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

n = 3 All non-symmetric parameters on these curves have non-conjugate dynamics. 3.If n > 2, there are uncountably many simple closed curves surrounding the McMullen domain; all these parameters are (non-escape) Sierpinski curves.

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 (j > 1).

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 (j > 1).

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 (j > 1).

n = 3 passes through 1,594,324 centers. 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 (j > 1).

As before, all non- symmetrically located centers have different dynamics. 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 (j > 1).

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

4.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.When

, the Julia set again expodes.When

, the Julia set again expodes.When

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

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

The case n = 2 is very different from (and much more difficult than) the case n > 2. n = 3 n = 2

One difference: there is a McMullen domain when n > 2, but no McMullen domain when n = 2 n = 3 n = 2

One difference: there is a McMullen domain when n > 2, but no McMullen domain when n = 2 n = 3 n = 2

There is lots of structure when n > 2, but what is going on when n = 2? n = 3 n = 2

There is lots of structure when n > 2, but what is going on when n = 2? n = 3 n = 2

There is lots of structure when n > 2, but what is going on when n = 2? n = 3 n = 2

Another difference: as n = 3 n = 2 when n > 2 when n = 2

Also, not much is happening for the Julia sets near when n > 2 n = 3

The Julia set is always a Cantor set of circles. n = 3

The Julia set is always a Cantor set of circles.

The Julia set is always a Cantor set of circles. There is always a round annulus of some fixed width in the Fatou set, so the Fatou set is “large.”

n = 2 But when n = 2, lots of things happen near the origin; in fact, the Julia sets converge to the unit disk as disk-converge

Here’s the parameter plane when n = 2:

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