Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno.

Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno Rocha Matt HolzerKevin Pilgrim U. Hoomiforgot Elizabeth Russell Dan LookYakov Shapiro Sebastian MarottaDavid Uminsky with:

Three different types of topological objects: 1. Cantor Necklaces A Cantor necklace is a planar set that is homeomorphic to the Cantor middle thirds set with open disks replacing removed intervals.

Three different types of topological objects: 1. Cantor Necklaces dynamical plane n = 2

Three different types of topological objects: 1. Cantor Necklaces dynamical plane parameter plane n = 2

Three different types of topological objects: 2. Mandelpinski Necklaces Infinitely many simple closed curves in the parameter plane that pass alternately through centers of “Sierpinski holes” and centers of baby Mandelbrot sets.

Three different types of topological objects: 2. Mandelpinski Necklaces parameter planezoom in n = 3

Three different types of topological objects: 3. CanManPinski Trees A tree of Cantor necklaces with Mandelbrot sets interspersed between each branch

Three different types of topological objects: 3. CanManPinski Trees parameter plane n = 2

Dynamics of complex and 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

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

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

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 But really only 1 free critical orbit since the map has 2n-fold symmetry

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

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) (with Dan Look & David Uminsky)

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

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 lies in a Sierpinski hole Case 3: the critical orbit eventually lands in the trap door.

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

1.Cantor necklaces in the dynamical and parameter plane The Cantor necklace is homeomorphic to the Cantor middle thirds set with open disks replacing removed intervals.

1.Cantor necklaces in the dynamical and parameter plane Julia set n = 2 = -0.23 The Cantor necklace is homeomorphic to the Cantor middle thirds set with open disks replacing removed intervals.

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

Dynamical plane: n = 2 Suppose B and T are disjoint. B T

Dynamical plane: n = 2 Four critical points 1/4

Dynamical plane: n = 2 And two critical values that do not lie in T 2 1/2

Dynamical plane: n = 2 The critical lines...

Dynamical plane: n = 2 are mapped two-to-one to one of two critical value rays

So the sectors S 0 and S 1 are mapped one-to-one to C - {critical value rays) S1S1 S0S0

And the regions I 0 - T and I 1 - T are mapped one-to-one to C - B - {critical value rays) Dynamical plane: n = 2 I1I1 I0I0

And the regions I 0 - T and I 1 - T are mapped one-to-one to C - B - {critical value rays) Dynamical plane: n = 2

I1I1 I0I0 So consider the bow-tie I 0  T  I 1 T

Dynamical plane: n = 2 I1I1 I0I0 Both I 0 and I 1 are mapped one-to-one over the entire bow-tie I 0  T  I 1 T

Dynamical plane: n = 2 So we have a preimage of the bow-tie inside each of I 0 and I 1 T

Dynamical plane: n = 2 Then a second preimage, etc., etc. T

Dynamical plane: n = 2 The points whose orbits stay in I 0  I 1 form a Cantor set, and the preimages of T give the adjoined disks. T

Dynamical plane: n = 2 The points whose orbits stay in I 0  I 1 form a Cantor set, and the preimages of T give the adjoined disks.

Cantor Necklaces in the Parameter Plane c = 1/4 v = 2 1/2 F (v ) = 1/4 + 4 D = { | | | < 1, Re( ) < 0} D

For each  D, have a Cantor set of points inside I 1 I1I1 I0I0 T........:...

I1I1 I0I0 T For each  D, have a Cantor set of points inside I 1........:... Let z s ( ) be the point in the Cantor set with itinerary s z s ( )

I1I1 I0I0 T For each  D, have a Cantor set of points inside I 1........:... Let z s ( ) be the point in the Cantor set with itinerary s z s ( ) z s ( ) depends analytically on and continuously on s

I1I1 I0I0 T For each  D, have a Cantor set of points inside I 1........:... Let z s ( ) be the point in the Cantor set with itinerary s z s ( ) and z s ( ) lies in the half-disk H given by |z| < 2, Re(z) < 0 H z s ( ) depends analytically on and continuously on s

So have an analytic map  z s ( ) that takes D into H z s ( ) D H

So have an analytic map  z s ( ) that takes D into H z s ( ) D H Have another map  G( ) = F (v ) = 1/4 + 4 which maps D over a larger half disk containing H G( )

So have an analytic map  z s ( ) that takes D into H z s ( ) D H Have another map  G( ) = F (v ) = 1/4 + 4 which maps D over a larger half disk containing H G -1 But G is invertible.

So have an analytic map  z s ( ) that takes D into H z s ( ) D H Have another map  G( ) = F (v ) = 1/4 + 4 which maps D over a larger half disk containing H G -1 But G is invertible. So G -1 (z s ( )) maps D strictly inside itself.

z s ( ) D H G -1 By the Schwarz Lemma, for each itinerary s there is a unique fixed point s for the map G -1 (z s ( )). s

z s ( ) D H This is a parameter for which G( s ) = z s ( s ), i.e., the second iterate of the critical points lands on a point in the Cantor set portion of the Cantor necklace. G -1 By the Schwarz Lemma, for each itinerary s there is a unique fixed point s for the map G -1 (z s ( )). s z s ( s )

z s ( ) D H G -1 So the points s for each s give a Cantor set of points in the parameter plane. s z s ( s )

z s ( ) D H G -1 So the points s for each s give a Cantor set of points in the parameter plane. s Similar arguments involving Böttcher coordinates on and itineraries of preimages of the trap door then append the Sierpinski holes to the necklace. z s ( s )

This necklace lies along the negative real axis. parameter plane n = 2

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

When n > 2, get more complicated Cantor webs case n = 3:

When n > 2, get more complicated Cantor webs case n = 3: Continue in this way and then adjoin Cantor sets

Parameter plane n = 3 Dynamical plane

Parameter plane n = 3 Cantor webs in the parameter plane

Parameter plane n = 3 Cantor webs in the parameter plane Parameter plane

Parameter plane n = 4 Different Cantor webs when n = 4 Dynamical plane

Part 2: Mandelpinski Necklaces

Parameter plane for n = 3 A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centers of baby Mandelbrot sets and k centers of Sierpinski holes.

Parameter plane for n = 3 A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centers of baby Mandelbrot sets and k centers of Sierpinski holes. C 1 passes through the centers of 2 M-sets and 2 S-holes Easy check: C 1 is the circle r = 2 -2n/n-1

Parameter plane for n = 3 A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centers of baby Mandelbrot sets and k centers of Sierpinski holes.

Parameter plane for n = 3 A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centers of baby Mandelbrot sets and k centers of Sierpinski holes. C 2 passes through the centers of 4 M-sets and 4 S-holes * * only exception: 2 centers of period 2 bulbs, not M-sets

A “Mandelpinski necklace” is a simple closed curve in the parameter plane that passes alternately through k centers of baby Mandelbrot sets and k centers of Sierpinski holes. Parameter plane for n = 3 C 3 passes through the centers of 10 M-sets and 10 S-holes

Theorem: There exist closed curves C j, surrounding the McMullen domain. Each C j passes alternately through (n-2)n j-1 +1 centers of baby Mandelbrot sets and centers of Sierpinski holes.

C 14 passes through the centers of 4,782,969 M-sets and S-holes Parameter plane for n = 3 Theorem: There exist closed curves C j, surrounding the McMullen domain. Each C j passes alternately through (n-2)n j-1 +1 centers of baby Mandelbrot sets and centers of Sierpinski holes.

Parameter plane for n = 4 C 1 : 3 holes and M-sets Theorem: There exist closed curves C j, surrounding the McMullen domain. Each C j passes alternately through (n-2)n j-1 +1 centers of baby Mandelbrot sets and centers of Sierpinski holes.

Parameter plane for n = 4 C 2 : 9 holes and M-sets C 3 : 33 holes and M-sets Theorem: There exist closed curves C j, surrounding the McMullen domain. Each C j passes alternately through (n-2)n j-1 +1 centers of baby Mandelbrot sets and centers of Sierpinski holes. *

Easy computations: Critical points: 1/2n Prepoles: (- ) 1/2n

Easy computations: All of the critical points and prepoles lie on the “critical circle” : |z| = | | 1/2n

All of the critical points and prepoles lie on the “critical circle” : |z| = | | 1/2n which is mapped 2n-to-1 onto the “critical value line” connecting Easy computations:

Any other circle around 0 is mapped n-to-1 to an ellipse whose foci are Easy computations:

Any other circle around 0 is mapped n-to-1 to an ellipse whose foci are

So the exterior of is mapped as an n-to-1 covering of the exterior of the critical value line. Easy computations: Any other circle around 0 is mapped n-to-1 to an ellipse whose foci are

So the exterior of is mapped as an n-to-1 covering of the exterior of the critical value line. Same with the interior of. Easy computations: Any other circle around 0 is mapped n-to-1 to an ellipse whose foci are

Now assume that lies inside the critical circle: Warning: this is not a real proof....

Now assume that lies inside the critical circle: The exterior of is mapped n-to-1 onto the exterior of the critical value ray, so there is a preimage mapped n-to-1 to,

then is mapped n-to-1 to, Now assume that lies inside the critical circle: The exterior of is mapped n-to-1 onto the exterior of the critical value ray, so there is a preimage mapped n-to-1 to,

and on out to Now assume that lies inside the critical circle: then is mapped n-to-1 to, B The exterior of is mapped n-to-1 onto the exterior of the critical value ray, so there is a preimage mapped n-to-1 to,

contains 2n critical points and 2n prepoles, so contains 2n 2 pre-critical points and pre-prepoles

contains 2n critical points and 2n prepoles, so contains 2n 2 pre-critical points and pre-prepoles B contains 2n k+1 points that map to the critical points and pre-prepoles under

contains 2n critical points and 2n prepoles, so contains 2n 2 pre-critical points and pre-prepoles contains 2n k+1 points that map to the critical points and pre-prepoles under n = 3

As rotates by one turn, these 2n k+1 points on each rotate by 1/2n k+1 of a turn.

Since the second iterate of the critical points rotate by 1 - n/2 of a turn As rotates by one turn, these 2n k+1 points on each rotate by 1/2n k+1 of a turn.

Since the second iterate of the critical points rotate by 1 - n/2 of a turn, so this point hits exactly preimages of the critical points and prepoles on As rotates by one turn, these 2n k+1 points on each rotate by 1/2n k+1 of a turn.

There is a natural parametrization of each The real proof involves the Schwarz Lemma (as before):

There is a natural parametrization of each The real proof involves the Schwarz Lemma (as before): Best to restrict to a “symmetry region” inside the circle C 1, so that is well-defined.

Best to restrict to a “symmetry region” inside the circle C 1, so that is well-defined. Then we have a second map from the parameter plane to the dynamical plane, namely which is invertible on the symmetry sector

Then we have a second map from the parameter plane to the dynamical plane, namely which is invertible on the symmetry sector a map from a “disk” to itself. So consider the composition

a map from a “disk” to itself. So consider the composition Schwarz implies that has a unique fixed point, i.e., a parameter for which the second iterate of the critical point lands on the point, so this proves the existence of lots of parameters for which the critical orbits are periodic and land on 0.

Remarks: 1. This proves the existence of centers of Sierpinski holes and Mandelbrot sets. Producing the entire M-set involves polynomial-like maps; while the entire S-hole involves qc-surgery.

Remarks: 1. This proves the existence of centers of Sierpinski holes and Mandelbrot sets. Producing the entire M-set involves polynomial-like maps; while the entire S-hole involves qc-surgery. 2.It is known that each S-hole in the Mandelpinski necklace is also surrounded by infinitely many sub-necklaces, which in turn are surrounded by sub-sub-necklaces, etc.

3. CanManPinski Trees A tree of Cantor necklaces with Mandelbrot sets interspersed between each branch parameter plane n = 2

Dynamical plane: n = 2 I1I1 I0I0 Recall that we have a Cantor necklace in the dynamical plane lying in I 0  T  I 1

Dynamical plane: n = 2 I1I1 I0I0 The regions I 2 and I 3 are mapped one-to-one over I 0  T  I 1, so there are Cantor necklaces in I 2 and I 3 I2I2 I3I3

Dynamical plane: n = 2 I1I1 I0I0 I2I2 I3I3 The regions I 2 and I 3 are mapped one-to-one over I 0  T  I 1, so there are Cantor necklaces in I 2 and I 3

This necklace is mapped one-to-one onto the original necklace.

This necklace is mapped one-to-one onto the original necklace. And so is the bottom necklace

S2S2 S0S0 S1S1 S3S3 Now consider the regions S j.

S2S2 S0S0 S1S1 S3S3 S 0 is mapped two-to-one onto S 0  S 1

S2S2 S0S0 S1S1 S3S3 Now consider the regions S j. S 0 is mapped two-to-one onto S 0  S 1 Similarly, S 1  S 2  S 3, S 2  S 0  S 1 and S 3  S 2  S 3

S2S2 S0S0 S1S1 S3S3 Assuming lies in the upper half plane, the critical values  v lie in S 0 and S 2 (easy check) v -v

S2S2 S0S0 S1S1 S3S3 Assuming lies in the upper half plane, the critical values  v lie in S 0 and S 2 (easy check) So there is a region in S 3 mapped one-to-one onto S 3. v -v

S2S2 S0S0 S1S1 S3S3 So there is a preimage of this Cantor necklace in S 3 v -v

S2S2 S0S0 S1S1 S3S3 So there is a preimage of this Cantor necklace in S 3, v -v

S2S2 S0S0 S1S1 S3S3 So there is a preimage of this Cantor necklace in S 3, and then another preimage, v -v

S2S2 S0S0 S1S1 S3S3 So there is a preimage of this Cantor necklace in S 3, and then another preimage, and so on, yielding infinitely many necklaces eventually mapping to the original necklace. Looking like branches of a tree.... v -v

S2S2 S0S0 S1S1 S3S3 By symmetry, we have similar branches in S 1 v -v

S2S2 S0S0 S1S1 S3S3 By symmetry, we have similar branches in S 1, S 0,

S2S2 S0S0 S1S1 S3S3 By symmetry, we have similar branches in S 1, S 0, and S 2

This produces trees of Cantor necklaces in the dynamical plane

Assuming is in the upper half plane, we can again use G( ) = 1/4 + 4 and an appropriate coding of points in the necklace, and then the Schwarz Lemma produces a similar tree in the upper half of the parameter plane.

Symmetry under complex conjugation yields a similar tree in the lower half-plane.

Then polynomial-like map theory produces a Mandelbrot set in each region in between the branches.

Open problems: Can we use these trees to give a complete map of the Sierpinski-hole regions and buried parameters in the parameter planes? Same question for the baby Mandelbrot sets. Using “Yoccoz puzzles,” do these trees allow us to see that the boundary of the parameter plane locus is a simple closed curve?

Open problems: Can we use these trees to give a complete map of the Sierpinski-hole regions and buried parameters in the parameter planes? Same question for the baby Mandelbrot sets. Using “Yoccoz puzzles,” do these trees allow us to see that the boundary of the parameter plane locus is a simple closed curve? Would it be better to call these things Cantormandelbrotsierpinski trees?

Open problems: Can we use these trees to give a complete map of the Sierpinski-hole regions and buried parameters in the parameter planes? Same question for the baby Mandelbrot sets. Using “Yoccoz puzzles,” do these trees allow us to see that the boundary of the parameter plane locus is a simple closed curve? Would it be better to call these things Cantormandelbrotsierpinski trees? Who the hell is this?

Parameter plane (rotated) when n = 2

Other topics: Main cardioid of a buried baby M-set Perturbed rabbit Convergence to the unit disk Major application

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 A Sierpinski curve, but very different dynamically from the earlier ones. 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.

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

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 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 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

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

Also, not much is happening for the Julia sets near 0 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 Julia set does not converge to the unit disk.

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.....

Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno.

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Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno.

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Presentation on theme: "Topological Structures in the Julia Sets of Rational Maps Dynamics of the family of complex maps Paul Blanchard Mark Morabito Toni Garijo Monica Moreno."— Presentation transcript:

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