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Exponential Dynamics and (Crazy) Topology. Cantor bouquets. Indecomposable continua. Exponential Dynamics and (Crazy) Topology. Cantor bouquets. Indecomposable continua. These are Julia sets of. Example 1: Cantor Bouquets. with Clara Bodelon Michael Hayes Gareth Roberts

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slide1

Exponential Dynamics and (Crazy) Topology

Cantor bouquets

Indecomposable continua

slide2

Exponential Dynamics and (Crazy) Topology

Cantor bouquets

Indecomposable continua

These are Julia sets of

slide3

Example 1: Cantor Bouquets

with

Clara Bodelon

Michael Hayes

Gareth Roberts

Ranjit Bhattacharjee

Lee DeVille

Monica Moreno Rocha

Kreso Josic

Alex Frumosu

Eileen Lee

slide4

Orbit of z:

Question: What is the fate of orbits?

slide5

Julia set of

*

J = closure of {orbits that escape to }

= closure {repelling periodic orbits}

= {chaotic set}

Fatou set

= complement of J

= predictable set

* not the boundary of {orbits that escape to }

slide6

For polynomials, it was the orbit of the

critical points that determined everything.

But has no critical points.

slide7

For polynomials, it was the orbit of the

critical points that determined everything.

But has no critical points.

But 0 is an asymptotic value; any far

left half-plane is wrapped infinitely often

around 0, just like a critical value.

So the orbit of 0 for the exponential plays

a similar role as in the quadratic family

(only what happens to the Julia sets

is very different in this case).

slide8

Example 1:

is a “Cantor bouquet”

slide9

Example 1:

is a “Cantor bouquet”

slide10

Example 1:

is a “Cantor bouquet”

attracting

fixed point

q

slide11

Example 1:

is a “Cantor bouquet”

The orbit of 0 always

tends this attracting

fixed point

attracting

fixed point

q

slide12

Example 1:

is a “Cantor bouquet”

q

p

repelling

fixed point

slide13

Example 1:

is a “Cantor bouquet”

q

x0

p

slide14

Example 1:

is a “Cantor bouquet”

And

for all

q

x0

p

slide17

So where is J?

in this half plane

slide18

So where is J?

Green points lie in the Fatou set

slide19

So where is J?

Green points lie in the Fatou set

slide20

So where is J?

Green points lie in the Fatou set

slide21

So where is J?

Green points lie in the Fatou set

slide22

So where is J?

Green points lie in the Fatou set

slide23

The Julia set is a collection of curves (hairs) in the right half plane, each with an endpoint

and a stem.

hairs

endpoints

stems

slide25

Colored points escape to

and so now are in the Julia set.

q

p

slide30

One such hair lies on the real axis.

repelling

fixed point

stem

slide33

So bounded orbits lie in the set of endpoints.

Repelling cycles lie

in the set of endpoints.

hairs

slide34

So bounded orbits lie in the set of endpoints.

Repelling cycles lie

in the set of endpoints.

hairs

So the endpoints are

dense in the bouquet.

slide35

So bounded orbits lie in the set of endpoints.

Repelling cycles lie

in the set of endpoints.

hairs

So the endpoints are

dense in the bouquet.

slide36

S

Some Facts:

slide37

S

Some Facts:

The only accessible points in J from

the Fatou set are the endpoints;

you cannot touch the stems

slide38

S

Some Crazy Facts:

The only accessible points in J from

the Fatou set are the endpoints;

you cannot touch the stems

The set of endpoints together with the

point at infinity is connected ...

slide39

S

Some Crazy Facts:

The only accessible points in J from

the Fatou set are the endpoints;

you cannot touch the stems

The set of endpoints together with the

point at infinity is connected ...

but the set of endpoints is

totally disconnected (Mayer)

slide40

S

Some Crazy Facts:

The only accessible points in J from

the Fatou set are the endpoints;

you cannot touch the stems

The set of endpoints together with the

point at infinity is connected ...

but the set of endpoints is

totally disconnected (Mayer)

Hausdorff dimension of {stems} = 1...

slide41

S

Some Crazy Facts:

The only accessible points in J from

the Fatou set are the endpoints;

you cannot touch the stems

The set of endpoints together with the

point at infinity is connected ...

but the set of endpoints is

totally disconnected (Mayer)

Hausdorff dimension of {stems} = 1...

but the Hausdorff dimension of

{endpoints} = 2! (Karpinska)

slide42

Another example:

Looks a little different, but still a pair

of Cantor bouquets

slide43

Another example:

The interval [-, ] is

contracted inside itself,

and all these orbits tend to

0 (so are in the Fatou set)

slide44

Another example:

The real line is contracted

onto the interval ,

and all these orbits tend to 0

(so are in the Fatou set)

slide45

Another example:

-/2

/2

The vertical lines x = n + /2 are

mapped to either [, ∞) or (-∞, - ],

so these lines are in the Fatou set....

slide46

Another example:

c

-c

The lines y = c are both

wrapped around an ellipse

with foci at

slide47

Another example:

c

-c

The lines y = c are both

wrapped around an ellipse

with foci at , and all orbits

in the ellipse tend to 0 if

c is small enough

slide48

Another example:

c

-c

So all points in the ellipse

lie in the Fatou set

slide49

Another example:

c

-c

So do all points in the

strip

slide50

Another example:

c

-c

The vertical lines given

by x = n + /2 are

also in the Fatou set.

slide51

And all points in the

preimages of the strip

lie in the Fatou set...

slide53

The difference here is that the Cantor bouquet for

the sine function has infinite Lebesgue measure,

while the exponential bouquet has zero measure.

slide54

Example 2: Indecomposable Continua

with

Nuria Fagella

Xavier Jarque

Monica Moreno Rocha

slide55

When

,

undergoes a “saddle node” bifurcation,

The two fixed points coalesce

and disappear from the real

axis when  goes above 1/e.

slide282

Theorem: If the orbit of 0 goes to ∞, then the

Julia set is the entire complex plane.

(Sullivan, Goldberg, Keen)

slide283

As

increases through

,

; however:

slide284

As

increases through

,

; however:

No new periodic cycles are born;

slide285

As

increases through

,

; however:

No new periodic cycles are born;

All move continuously to fill in the plane densely;

slide286

As

increases through

,

; however:

No new periodic cycles are born;

All move continuously to fill in the plane;

Infinitely many hairs suddenly become

“indecomposable continua.”

slide287

An indecomposable continuum is a compact, connected

set that cannot be broken into the union of two (proper)

compact, connected subsets.

For example:

slide288

An indecomposable continuum is a compact, connected

set that cannot be broken into the union of two (proper)

compact, connected subsets.

For example: indecomposable?

0

1

slide289

An indecomposable continuum is a compact, connected

set that cannot be broken into the union of two (proper)

compact, connected subsets.

No, decomposable.

For example:

0

1

(subsets need not be disjoint)

slide290

An indecomposable continuum is a compact, connected

set that cannot be broken into the union of two (proper)

compact, connected subsets.

For example: indecomposable?

slide291

An indecomposable continuum is a compact, connected

set that cannot be broken into the union of two (proper)

compact, connected subsets.

No, decomposable.

For example:

slide292

An indecomposable continuum is a compact, connected

set that cannot be broken into the union of two (proper)

compact, connected subsets.

For example: indecomposable?

slide293

An indecomposable continuum is a compact, connected

set that cannot be broken into the union of two (proper)

compact, connected subsets.

No, decomposable.

For example:

slide294

Knaster continuum

A well known example of an indecomposable continuum

Start with the Cantor middle-thirds set

slide295

Knaster continuum

Connect symmetric points about 1/2 with semicircles

slide296

Knaster continuum

Do the same below about 5/6

slide297

Knaster continuum

And continue....

slide299

Properties of K:

There is one curve that

passes through all the

endpoints of the Cantor

set.

slide300

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.

slide301

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.

slide302

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.

slide303

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.

So K is compact, connected, and....

slide304

Indecomposable!

Try to write K as the union

of two compact, connected sets.

slide305

Indecomposable!

Can’t divide it this way....

subsets are closed

but not connected.

slide306

Indecomposable!

Or this way...

again closed but

not connected.

slide307

Indecomposable!

Or the union of the outer curve and all

the inaccessible curves ... not closed.

slide308

How the hairs become indecomposable:

repelling

fixed pt

.

.

.

.

... .

.

.

.

.

attracting

fixed pt

stem

slide309

How the hairs become indecomposable:

.

.

.

.

... .

.

.

.

.

.

.

.

.

2 repelling

fixed points

.

.

.

.

.

.

.

.

Now all points in R escape,

so the hair is much longer

.

.

.

.

slide316

Compactify to get a single curve in a compact region

in the plane that accumulates everywhere on itself.

The closure is then an indecomposable continuum.

0

slide317

The dynamics on this continuum is very simple:

one repelling fixed point

all other orbits either tend to

or accumulate on the orbit of 0 and

But the topology is not at all understood:

Conjecture: the continuum for each

parameter is topologically distinct.

sin(z)

slide319

A pair of Cantor bouquets

Julia set of sin(z)

slide320

A pair of Cantor bouquets

A similar explosion occurs for

the sine family (1 + ci) sin(z)

Julia set of sin(z)

slide526

Questions:

Do the hairs become indecomposable

continua as in the exponential case?

If so, what is the topology of these sets?

slide527

Parameter plane for

To plot the parameter plane (the analogue

of the Mandelbrot set), for each  plot the

corresponding orbit of 0.

If 0 escapes, the color ; J is the entire plane.

If 0 does not escape, leave  black; J is

usually a “pinched” Cantor bouquet.

slide531

Parameter plane for

has an

attracting fixed

point in this

cardioid

1

slide532

Parameter plane for

2

Period 2 region

1

slide534

Fixed point bifurcations

2

Period 2 region

1

So undergoes

a period doubling

bifurcation along

this path in the

parameter plane

at

Parameter plane for

slide536

The Cantor bouquet

A repelling

2-cycle at

two endpoints

slide557

The hairs containing the 2-cycle meet at the

neutral fixed point, and then remain attached.

Meanwhile an attracting 2 cycle emerges.

We get a “pinched” Cantor bouquet

slide558

Other bifurcations

Period 3 region

3

2

1

Parameter plane for

slide559

Other bifurcations

Period tripling

bifurcation

3

2

1

Parameter plane for

slide560

Three hairs containing a 3-cycle meet at the

neutral fixed point, and then remain attached

We get a different “pinched” Cantor bouquet

slide561

Other bifurcations

Period 5

bifurcation

5

3

2

1

Parameter plane for

slide562

Five hairs containing a 5-cycle meet at the

neutral fixed point, and then remain attached

We get a different “pinched” Cantor bouquet

slide565

4

1

5

slide566

On a path like this, we pass through infinitely

many regions where there is an attracting cycle,

so J is a pinched Cantor bouquet.....

and infinitely many hairs where J is the entire plane.

slide638

Example 3: Sierpinski Curves

with:

Paul Blanchard

Toni Garijo

Matt Holzer

U. Hoomiforgot

Dan Look

Sebastian Marotta

Mark Morabito

Monica Moreno Rocha

Kevin Pilgrim

Elizabeth Russell

Yakov Shapiro

David Uminsky

slide639

Sierpinski Curve

A Sierpinski curve is any planar

set that is homeomorphic to the Sierpinski carpet fractal.

The Sierpinski Carpet

slide640

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

slide641

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....

slide642

can be embedded inside

The topologist’s sine curve

slide643

can be embedded inside

The topologist’s sine curve

slide644

can be embedded inside

The topologist’s sine curve

slide645

can be embedded inside

The Knaster continuum

slide646

can be embedded inside

The Knaster continuum

slide647

can be embedded inside

The Knaster continuum

slide648

can be embedded inside

The Knaster continuum

slide649

can be embedded inside

The Knaster continuum

slide650

can be embedded inside

The Knaster continuum

slide651

can be embedded inside

The Knaster continuum

slide652

can be embedded inside

The Knaster continuum

slide653

can be embedded inside

The Knaster continuum

slide654

can be embedded inside

The Knaster continuum

slide655

can be embedded inside

The Knaster continuum

slide656

can be embedded inside

The Knaster continuum

slide657

can be embedded inside

The Knaster continuum

slide658

can be embedded inside

The Knaster continuum

slide659

can be embedded inside

The Knaster continuum

slide660

can be embedded inside

The Knaster continuum

slide661

can be embedded inside

The Knaster continuum

slide662

can be embedded inside

The Knaster continuum

slide663

can be embedded inside

The Knaster continuum

slide664

can be embedded inside

The Knaster continuum

slide665

can be embedded inside

The Knaster continuum

slide666

can be embedded inside

The Knaster continuum

slide667

can be embedded inside

The Knaster continuum

slide668

can be embedded inside

The Knaster continuum

slide669

can be embedded inside

The Knaster continuum

slide670

can be embedded inside

The Knaster continuum

slide671

can be embedded inside

The Knaster continuum

slide672

can be embedded inside

The Knaster continuum

slide673

can be embedded inside

The Knaster continuum

slide674

can be embedded inside

The Knaster continuum

slide675

can be embedded inside

The Knaster continuum

slide676

can be embedded inside

The Knaster continuum

slide677

can be embedded inside

The Knaster continuum

slide678

can be embedded inside

The Knaster continuum

slide679

can be embedded inside

Even this “curve”

slide681

Some easy to verify facts:

Have an immediate basin of infinity B

slide682

Some easy to verify facts:

Have an immediate basin of infinity B

0 is a pole so have a “trap door” T

(the preimage of B)

slide683

Some easy to verify facts:

Have an immediate basin of infinity B

0 is a pole so have a “trap door” T

(the preimage of B)

2n critical points given by but really only

one critical orbit due to symmetry

slide684

Some easy to verify facts:

Have an immediate basin of infinity B

0 is a pole so have a “trap door” T

(the preimage of B)

2n critical points given by but really only

one critical orbit due to symmetry

J is now the boundary of the escaping orbits

(not the closure)

slide685

When , the Julia set

is the unit circle

slide686

When , the Julia set

is the unit circle

But when , the

Julia set explodes

A Sierpinskicurve

slide687

When , the Julia set

is the unit circle

But when , the

Julia set explodes

B

T

A Sierpinskicurve

slide688

When , the Julia set

is the unit circle

But when , the

Julia set explodes

Another Sierpinskicurve

slide689

When , the Julia set

is the unit circle

But when , the

Julia set explodes

Also a Sierpinski curve

slide690

Sierpinski curves arise in lots of different

ways in these families:

1. If the critical orbits

eventually fall into

the trap door (which

is disjoint from B),

then J is a Sierpinski

curve.

slide691

Sierpinski curves arise in lots of different

ways in these families:

1. If the critical orbits

eventually fall into

the trap door (which

is disjoint from B),

then J is a Sierpinski

curve.

slide692

Sierpinski curves arise in lots of different

ways in these families:

1. If the critical orbits

eventually fall into

the trap door (which

is disjoint from B),

then J is a Sierpinski

curve.

slide693

Sierpinski curves arise in lots of different

ways in these families:

1. If the critical orbits

eventually fall into

the trap door (which

is disjoint from B),

then J is a Sierpinski

curve.

slide694

Sierpinski curves arise in lots of different

ways in these families:

1. If the critical orbits

eventually fall into

the trap door (which

is disjoint from B),

then J is a Sierpinski

curve.

slide695

Sierpinski curves arise in lots of different

ways in these families:

1. If the critical orbits

eventually fall into

the trap door (which

is disjoint from B),

then J is a Sierpinski

curve.

slide696

Sierpinski curves arise in lots of different

ways in these families:

1. If the critical orbits

eventually fall into

the trap door (which

is disjoint from B),

then J is a Sierpinski

curve.

slide697

Sierpinski curves arise in lots of different

ways in these families:

1. If the critical orbits

eventually fall into

the trap door (which

is disjoint from B),

then J is a Sierpinski

curve.

slide698

Sierpinski curves arise in lots of different

ways in these families:

1. If the critical orbits

eventually fall into

the trap door (which

is disjoint from B),

then J is a Sierpinski

curve.

slide699

Sierpinski curves arise in lots of different

ways in these families:

1. If the critical orbits

eventually fall into

the trap door (which

is disjoint from B),

then J is a Sierpinski

curve.

slide700

Sierpinski curves arise in lots of different

ways in these families:

1. If the critical orbits

eventually fall into

the trap door (which

is disjoint from B),

then J is a Sierpinski

curve.

slide701

Lots of ways this happens:

parameter plane

when n = 3

slide702

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole

slide703

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole

slide704

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole

slide705

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole

slide706

Theorem: Two maps drawn from the same Sierpinski

hole have the same dynamics, but those drawn from

different holes are not conjugate (except in very

few symmetric cases).

n = 4, escape time 4, 24 Sierpinski holes,

slide707

Theorem: Two maps drawn from the same Sierpinski

hole have the same dynamics, but those drawn from

different holes are not conjugate (except in very

few symmetric cases).

n = 4, escape time 4, 24 Sierpinski holes,

but only five conjugacy classes

slide708

Theorem: Two maps drawn from the same Sierpinski

hole have the same dynamics, but those drawn from

different holes are not conjugate (except in very

few symmetric cases).

n = 4, escape time 12: 402,653,184 Sierpinski holes,

but only 67,108,832distinctconjugacy classes

Sorry. I forgot to

indicate their locations.

slide709

Sierpinski curves arise in lots of different

ways in these families:

2. If the parameter lies in

the main cardioid of a

buried baby Mandelbrot

set, J is again a

Sierpinski curve.

parameter plane

when n = 4

slide710

Sierpinski curves arise in lots of different

ways in these families:

2. If the parameter lies in

the main cardioid of a

buried baby Mandelbrot

set, J is again a

Sierpinski curve.

parameter plane

when n = 4

slide711

Sierpinski curves arise in lots of different

ways in these families:

2. If the parameter lies in

the main cardioid of a

buried baby Mandelbrot

set, J is again a

Sierpinski curve.

parameter plane

when n = 4

slide712

Sierpinski curves arise in lots of different

ways in these families:

2. If the parameter lies in

the main cardioid of a

buried baby Mandelbrot

set, J is again a

Sierpinski curve.

Black regions are the basin

of an attracting cycle.

slide713

Sierpinski curves arise in lots of different

ways in these families:

3. If the parameter lies

at a buried point in the

“Cantor necklaces” in

the parameter plane,

J is again a Sierpinski

curve.

parameter plane

n = 4

slide714

Sierpinski curves arise in lots of different

ways in these families:

3. If the parameter lies

at a buried point in the

“Cantor necklaces” in

the parameter plane,

J is again a Sierpinski

curve.

parameter plane

n = 4

slide715

Sierpinski curves arise in lots of different

ways in these families:

3. If the parameter lies

at a buried point in the

“Cantor necklaces” in

the parameter plane,

J is again a Sierpinski

curve.

parameter plane

n = 4

slide716

Sierpinski curves arise in lots of different

ways in these families:

4. There is a Cantor set

of circles in the parameter

plane on which each

parameter corresponds

to a Sierpinski curve.

n = 3

slide717

Sierpinski curves arise in lots of different

ways in these families:

4. There is a Cantor set

of circles in the parameter

plane on which each

parameter corresponds

to a Sierpinski curve.

n = 3

slide718

Sierpinski curves arise in lots of different

ways in these families:

4. There is a Cantor set

of circles in the parameter

plane on which each

parameter corresponds

to a Sierpinski curve.

n = 3

slide719

Theorem: All these Julia sets are the same topologically, but they are all (except for symmetrically located parameters) VERY different from a dynamics point of view (i.e., the maps are not conjugate).

Problem: Classify the dynamics on these

Sierpinski curve Julia sets.

slide721

Corollary:

Yes, those planar topologists

are crazy, but I sure wish I were one of them!

slide722

Corollary:

Yes, those planar topologists

are crazy, but I sure wish I were one of them!

The End!

slide723

website:

math.bu.edu/DYSYS

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