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Exponential Dynamics and (Crazy) Topology

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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

Ranjit Bhattacharjee

Lee DeVille

Monica Moreno Rocha

Kreso Josic

Alex Frumosu

Eileen Lee

Orbit of z:

Question: What is the fate of orbits?

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 }

For polynomials, it was the orbit of the

critical points that determined everything.

But has no critical points.

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).

Example 1:

is a “Cantor bouquet”

Example 1:

is a “Cantor bouquet”

Example 1:

is a “Cantor bouquet”

attracting

fixed point

q

Example 1:

is a “Cantor bouquet”

The orbit of 0 always

tends this attracting

fixed point

attracting

fixed point

q

Example 1:

is a “Cantor bouquet”

q

p

repelling

fixed point

Example 1:

is a “Cantor bouquet”

q

x0

p

Example 1:

is a “Cantor bouquet”

And

for all

q

x0

p

So where is J?

So where is J?

So where is J?

in this half plane

So where is J?

Green points lie in the Fatou set

So where is J?

Green points lie in the Fatou set

So where is J?

Green points lie in the Fatou set

So where is J?

Green points lie in the Fatou set

So where is J?

Green points lie in the Fatou set

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

and a stem.

hairs

endpoints

stems

A “Cantor bouquet”

q

p

Colored points escape to

and so now are in the Julia set.

q

p

One such hair lies on the real axis.

repelling

fixed point

stem

Orbits of points on the stems all tend to .

hairs

So bounded orbits lie in the set of endpoints.

hairs

So bounded orbits lie in the set of endpoints.

Repelling cycles lie

in the set of endpoints.

hairs

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.

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.

S

Some Facts:

S

Some Facts:

The only accessible points in J from

the Fatou set are the endpoints;

you cannot touch the stems

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 ...

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)

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...

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)

Another example:

Looks a little different, but still a pair

of Cantor bouquets

Another example:

The interval [-, ] is

contracted inside itself,

and all these orbits tend to

0 (so are in the Fatou set)

Another example:

The real line is contracted

onto the interval ,

and all these orbits tend to 0

(so are in the Fatou set)

Another example:

-/2

/2

The vertical lines x = n + /2 are

mapped to either [, ∞) or (-∞, - ],

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

Another example:

c

-c

The lines y = c are both

wrapped around an ellipse

with foci at

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

Another example:

c

-c

So all points in the ellipse

lie in the Fatou set

Another example:

c

-c

So do all points in the

strip

Another example:

c

-c

The vertical lines given

by x = n + /2 are

also in the Fatou set.

And all points in the

preimages of the strip

lie in the Fatou set...

And so on to get another

Cantor bouquet.

The difference here is that the Cantor bouquet for

the sine function has infinite Lebesgue measure,

while the exponential bouquet has zero measure.

Example 2: Indecomposable Continua

with

Nuria Fagella

Xavier Jarque

Monica Moreno Rocha

When

,

undergoes a “saddle node” bifurcation,

The two fixed points coalesce

and disappear from the real

axis when goes above 1/e.

And now the orbit of 0 goes off to ∞ ....

And as increases through 1/e, explodes.

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

Julia set is the entire complex plane.

(Sullivan, Goldberg, Keen)

As

increases through

,

; however:

As

increases through

,

; however:

No new periodic cycles are born;

As

increases through

,

; however:

No new periodic cycles are born;

All move continuously to fill in the plane densely;

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.”

An indecomposable continuum is a compact, connected

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

compact, connected subsets.

For example:

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

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)

An indecomposable continuum is a compact, connected

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

compact, connected subsets.

For example: indecomposable?

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:

An indecomposable continuum is a compact, connected

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

compact, connected subsets.

For example: indecomposable?

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:

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.

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....

Indecomposable!

Try to write K as the union

of two compact, connected sets.

Indecomposable!

Can’t divide it this way....

subsets are closed

but not connected.

Indecomposable!

Or this way...

again closed but

not connected.

Indecomposable!

Or the union of the outer curve and all

the inaccessible curves ... not closed.

How the hairs become indecomposable:

repelling

fixed pt

.

.

.

.

... .

.

.

.

.

attracting

fixed pt

stem

How the hairs become indecomposable:

.

.

.

.

... .

.

.

.

.

.

.

.

.

2 repelling

fixed points

.

.

.

.

.

.

.

.

Now all points in R escape,

so the hair is much longer

.

.

.

.

But the hair is even longer!

0

But the hair is even longer!

0

But the hair is even longer! And longer.

0

But the hair is even longer! And longer...

0

But the hair is even longer! And longer.......

0

But the hair is even longer! And longer.............

0

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

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)

A pair of Cantor bouquets

Julia set of sin(z)

A pair of Cantor bouquets

A similar explosion occurs for

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

Julia set of sin(z)

sin(z)

sin(z)

sin(z)

(1+.2i) sin(z)

(1+ ci) sin(z)

Questions:

Do the hairs become indecomposable

continua as in the exponential case?

If so, what is the topology of these sets?

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.

Parameter plane for

Parameter plane for

Parameter plane for

Parameter plane for

has an

attracting fixed

point in this

cardioid

1

Parameter plane for

2

Period 2 region

1

Parameter plane for

4

3

5

2

1

5

3

4

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

The Cantor bouquet

The Cantor bouquet

A repelling

2-cycle at

two endpoints

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

Other bifurcations

Period 3 region

3

2

1

Parameter plane for

Other bifurcations

Period tripling

bifurcation

3

2

1

Parameter plane for

Three hairs containing a 3-cycle meet at the

neutral fixed point, and then remain attached

We get a different “pinched” Cantor bouquet

Other bifurcations

Period 5

bifurcation

5

3

2

1

Parameter plane for

Five hairs containing a 5-cycle meet at the

neutral fixed point, and then remain attached

We get a different “pinched” Cantor bouquet

Lots of explosions occur....

3

1

Lots of explosions occur....

4

1

5

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.

slower

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

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

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....

can be embedded inside

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

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

The Knaster continuum

can be embedded inside

Even this “curve”

Some easy to verify facts:

Some easy to verify facts:

Have an immediate basin of infinity B

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)

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

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)

When , the Julia set

is the unit circle

When , the Julia set

is the unit circle

But when , the

Julia set explodes

A Sierpinskicurve

When , the Julia set

is the unit circle

But when , the

Julia set explodes

B

T

A Sierpinskicurve

When , the Julia set

is the unit circle

But when , the

Julia set explodes

Another Sierpinskicurve

When , the Julia set

is the unit circle

But when , the

Julia set explodes

Also a Sierpinski curve

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Lots of ways this happens:

parameter plane

when n = 3

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole

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,

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

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.

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

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

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

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.

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

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

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

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

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

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

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.

Corollary:

Corollary:

Yes, those planar topologists

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

Corollary:

Yes, those planar topologists

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

The End!

website:

math.bu.edu/DYSYS