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Exponential Dynamics and (Crazy) Topology

Cantor bouquets

Indecomposable continua

These are Julia sets of

with

Clara Bodelon

Michael Hayes

Gareth Roberts

Ranjit Bhattacharjee

Lee DeVille

Monica Moreno Rocha

Kreso Josic

Alex Frumosu

Eileen Lee

Question: What is the fate of orbits?

*

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).

is a “Cantor bouquet”

is a “Cantor bouquet”

is a “Cantor bouquet”

The orbit of 0 always

tends this attracting

fixed point

attracting

fixed point

q

in this half plane

Green points lie in the Fatou set

Green points lie in the Fatou set

Green points lie in the Fatou set

Green points lie in the Fatou set

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

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.

Some Facts:

Some Facts:

The only accessible points in J from

the Fatou set are the endpoints;

you cannot touch the stems

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 ...

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)

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...

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)

The interval [-, ] is

contracted inside itself,

and all these orbits tend to

0 (so are in the Fatou set)

The real line is contracted

onto the interval ,

and all these orbits tend to 0

(so are in the Fatou set)

-/2

/2

The vertical lines x = n + /2 are

mapped to either [, ∞) or (-∞, - ],

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

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

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.

,

undergoes a “saddle node” bifurcation,

The two fixed points coalesce

and disappear from the real

axis when goes above 1/e.

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

Julia set is the entire complex plane.

(Sullivan, Goldberg, Keen)

increases through

,

; however:

No new periodic cycles are born;

All move continuously to fill in the plane densely;

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:

A well known example of an indecomposable continuum

Start with the Cantor middle-thirds set

Connect symmetric points about 1/2 with semicircles

Do the same below about 5/6

And continue....

There is one curve that

passes through all the

endpoints of the Cantor

set.

It accumulates everywhere on

itself and on 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.

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.

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....

How the hairs become indecomposable:

.

.

.

.

... .

.

.

.

.

.

.

.

.

2 repelling

fixed points

.

.

.

.

.

.

.

.

Now all points in R escape,

so the hair is much longer

.

.

.

.

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)

Julia set of sin(z)

A similar explosion occurs for

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

Julia set of sin(z)

Do the hairs become indecomposable

continua as in the exponential case?

If so, what is the topology of these sets?

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.

2

Period 2 region

1

So undergoes

a period doubling

bifurcation along

this path in the

parameter plane

at

Parameter plane for

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

Three hairs containing a 3-cycle meet at the

neutral fixed point, and then remain attached

We get a different “pinched” Cantor bouquet

Five hairs containing a 5-cycle meet at the

neutral fixed point, and then remain attached

We get a different “pinched” Cantor bouquet

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.

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

A Sierpinski curve is any planar

set that is homeomorphic to the Sierpinski carpet fractal.

The Sierpinski Carpet

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:

Any planar, one-dimensional, compact, connected set can be homeomorphically embedded in a Sierpinski curve.

For example....

The topologist’s sine curve

The topologist’s sine curve

The topologist’s sine curve

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

The Knaster continuum

Even this “curve”

Have an immediate basin of infinity B

Have an immediate basin of infinity B

0 is a pole so have a “trap door” T

(the preimage of B)

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

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)

is the unit circle

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.

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole

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.

math.bu.edu/DYSYS

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