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

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Exponential dynamics and crazy topology

Exponential Dynamics and (Crazy) Topology

Cantor bouquets

Indecomposable continua


Exponential dynamics and crazy topology

Exponential Dynamics and (Crazy) Topology

Cantor bouquets

Indecomposable continua

These are Julia sets of


Exponential dynamics and crazy topology

Example 1: Cantor Bouquets

with

Clara Bodelon

Michael Hayes

Gareth Roberts

Ranjit Bhattacharjee

Lee DeVille

Monica Moreno Rocha

Kreso Josic

Alex Frumosu

Eileen Lee


Exponential dynamics and crazy topology

Orbit of z:

Question: What is the fate of orbits?


Exponential dynamics and crazy topology

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 }


Exponential dynamics and crazy topology

For polynomials, it was the orbit of the

critical points that determined everything.

But has no critical points.


Exponential dynamics and crazy topology

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).


Exponential dynamics and crazy topology

Example 1:

is a “Cantor bouquet”


Exponential dynamics and crazy topology

Example 1:

is a “Cantor bouquet”


Exponential dynamics and crazy topology

Example 1:

is a “Cantor bouquet”

attracting

fixed point

q


Exponential dynamics and crazy topology

Example 1:

is a “Cantor bouquet”

The orbit of 0 always

tends this attracting

fixed point

attracting

fixed point

q


Exponential dynamics and crazy topology

Example 1:

is a “Cantor bouquet”

q

p

repelling

fixed point


Exponential dynamics and crazy topology

Example 1:

is a “Cantor bouquet”

q

x0

p


Exponential dynamics and crazy topology

Example 1:

is a “Cantor bouquet”

And

for all

q

x0

p


Exponential dynamics and crazy topology

So where is J?


Exponential dynamics and crazy topology

So where is J?


Exponential dynamics and crazy topology

So where is J?

in this half plane


Exponential dynamics and crazy topology

So where is J?

Green points lie in the Fatou set


Exponential dynamics and crazy topology

So where is J?

Green points lie in the Fatou set


Exponential dynamics and crazy topology

So where is J?

Green points lie in the Fatou set


Exponential dynamics and crazy topology

So where is J?

Green points lie in the Fatou set


Exponential dynamics and crazy topology

So where is J?

Green points lie in the Fatou set


Exponential dynamics and crazy topology

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

and a stem.

hairs

endpoints

stems


Exponential dynamics and crazy topology

A “Cantor bouquet”

q

p


Exponential dynamics and crazy topology

Colored points escape to

and so now are in the Julia set.

q

p


Exponential dynamics and crazy topology

One such hair lies on the real axis.

repelling

fixed point

stem


Exponential dynamics and crazy topology

Orbits of points on the stems all tend to .

hairs


Exponential dynamics and crazy topology

So bounded orbits lie in the set of endpoints.

hairs


Exponential dynamics and crazy topology

So bounded orbits lie in the set of endpoints.

Repelling cycles lie

in the set of endpoints.

hairs


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

S

Some Facts:


Exponential dynamics and crazy topology

S

Some Facts:

The only accessible points in J from

the Fatou set are the endpoints;

you cannot touch the stems


Exponential dynamics and crazy topology

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 ...


Exponential dynamics and crazy topology

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)


Exponential dynamics and crazy topology

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...


Exponential dynamics and crazy topology

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)


Exponential dynamics and crazy topology

Another example:

Looks a little different, but still a pair

of Cantor bouquets


Exponential dynamics and crazy topology

Another example:

The interval [-, ] is

contracted inside itself,

and all these orbits tend to

0 (so are in the Fatou set)


Exponential dynamics and crazy topology

Another example:

The real line is contracted

onto the interval ,

and all these orbits tend to 0

(so are in the Fatou set)


Exponential dynamics and crazy topology

Another example:

-/2

/2

The vertical lines x = n + /2 are

mapped to either [, ∞) or (-∞, - ],

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


Exponential dynamics and crazy topology

Another example:

c

-c

The lines y = c are both

wrapped around an ellipse

with foci at


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

Another example:

c

-c

So all points in the ellipse

lie in the Fatou set


Exponential dynamics and crazy topology

Another example:

c

-c

So do all points in the

strip


Exponential dynamics and crazy topology

Another example:

c

-c

The vertical lines given

by x = n + /2 are

also in the Fatou set.


Exponential dynamics and crazy topology

And all points in the

preimages of the strip

lie in the Fatou set...


Exponential dynamics and crazy topology

And so on to get another

Cantor bouquet.


Exponential dynamics and crazy topology

The difference here is that the Cantor bouquet for

the sine function has infinite Lebesgue measure,

while the exponential bouquet has zero measure.


Exponential dynamics and crazy topology

Example 2: Indecomposable Continua

with

Nuria Fagella

Xavier Jarque

Monica Moreno Rocha


Exponential dynamics and crazy topology

When

,

undergoes a “saddle node” bifurcation,

The two fixed points coalesce

and disappear from the real

axis when  goes above 1/e.


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

And as increases through 1/e, explodes.


Exponential dynamics and crazy topology

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

Julia set is the entire complex plane.

(Sullivan, Goldberg, Keen)


Exponential dynamics and crazy topology

As

increases through

,

; however:


Exponential dynamics and crazy topology

As

increases through

,

; however:

No new periodic cycles are born;


Exponential dynamics and crazy topology

As

increases through

,

; however:

No new periodic cycles are born;

All move continuously to fill in the plane densely;


Exponential dynamics and crazy topology

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.”


Exponential dynamics and crazy topology

An indecomposable continuum is a compact, connected

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

compact, connected subsets.

For example:


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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)


Exponential dynamics and crazy topology

An indecomposable continuum is a compact, connected

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

compact, connected subsets.

For example: indecomposable?


Exponential dynamics and crazy topology

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:


Exponential dynamics and crazy topology

An indecomposable continuum is a compact, connected

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

compact, connected subsets.

For example: indecomposable?


Exponential dynamics and crazy topology

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:


Exponential dynamics and crazy topology

Knaster continuum

A well known example of an indecomposable continuum

Start with the Cantor middle-thirds set


Exponential dynamics and crazy topology

Knaster continuum

Connect symmetric points about 1/2 with semicircles


Exponential dynamics and crazy topology

Knaster continuum

Do the same below about 5/6


Exponential dynamics and crazy topology

Knaster continuum

And continue....


Exponential dynamics and crazy topology

Knaster continuum


Exponential dynamics and crazy topology

Properties of K:

There is one curve that

passes through all the

endpoints of the Cantor

set.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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....


Exponential dynamics and crazy topology

Indecomposable!

Try to write K as the union

of two compact, connected sets.


Exponential dynamics and crazy topology

Indecomposable!

Can’t divide it this way....

subsets are closed

but not connected.


Exponential dynamics and crazy topology

Indecomposable!

Or this way...

again closed but

not connected.


Exponential dynamics and crazy topology

Indecomposable!

Or the union of the outer curve and all

the inaccessible curves ... not closed.


Exponential dynamics and crazy topology

How the hairs become indecomposable:

repelling

fixed pt

.

.

.

.

... .

.

.

.

.

attracting

fixed pt

stem


Exponential dynamics and crazy topology

How the hairs become indecomposable:

.

.

.

.

... .

.

.

.

.

.

.

.

.

2 repelling

fixed points

.

.

.

.

.

.

.

.

Now all points in R escape,

so the hair is much longer

.

.

.

.


Exponential dynamics and crazy topology

But the hair is even longer!

0


Exponential dynamics and crazy topology

But the hair is even longer!

0


Exponential dynamics and crazy topology

But the hair is even longer! And longer.

0


Exponential dynamics and crazy topology

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

0


Exponential dynamics and crazy topology

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

0


Exponential dynamics and crazy topology

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

0


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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)


Exponential dynamics and crazy topology

A pair of Cantor bouquets

Julia set of sin(z)


Exponential dynamics and crazy topology

A pair of Cantor bouquets

A similar explosion occurs for

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

Julia set of sin(z)


Exponential dynamics and crazy topology

sin(z)


Exponential dynamics and crazy topology

sin(z)


Exponential dynamics and crazy topology

sin(z)


Exponential dynamics and crazy topology

(1+.2i) sin(z)


Exponential dynamics and crazy topology

(1+ ci) sin(z)


Exponential dynamics and crazy topology

Questions:

Do the hairs become indecomposable

continua as in the exponential case?

If so, what is the topology of these sets?


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

Parameter plane for


Exponential dynamics and crazy topology

Parameter plane for


Exponential dynamics and crazy topology

Parameter plane for


Exponential dynamics and crazy topology

Parameter plane for

has an

attracting fixed

point in this

cardioid

1


Exponential dynamics and crazy topology

Parameter plane for

2

Period 2 region

1


Exponential dynamics and crazy topology

Parameter plane for

4

3

5

2

1

5

3

4


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

The Cantor bouquet


Exponential dynamics and crazy topology

The Cantor bouquet

A repelling

2-cycle at

two endpoints


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

Other bifurcations

Period 3 region

3

2

1

Parameter plane for


Exponential dynamics and crazy topology

Other bifurcations

Period tripling

bifurcation

3

2

1

Parameter plane for


Exponential dynamics and crazy topology

Three hairs containing a 3-cycle meet at the

neutral fixed point, and then remain attached

We get a different “pinched” Cantor bouquet


Exponential dynamics and crazy topology

Other bifurcations

Period 5

bifurcation

5

3

2

1

Parameter plane for


Exponential dynamics and crazy topology

Five hairs containing a 5-cycle meet at the

neutral fixed point, and then remain attached

We get a different “pinched” Cantor bouquet


Exponential dynamics and crazy topology

Lots of explosions occur....

3

1


Exponential dynamics and crazy topology

Lots of explosions occur....


Exponential dynamics and crazy topology

4

1

5


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

slower


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

Sierpinski Curve

A Sierpinski curve is any planar

set that is homeomorphic to the Sierpinski carpet fractal.

The Sierpinski Carpet


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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....


Exponential dynamics and crazy topology

can be embedded inside

The topologist’s sine curve


Exponential dynamics and crazy topology

can be embedded inside

The topologist’s sine curve


Exponential dynamics and crazy topology

can be embedded inside

The topologist’s sine curve


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

The Knaster continuum


Exponential dynamics and crazy topology

can be embedded inside

Even this “curve”


Exponential dynamics and crazy topology

Some easy to verify facts:


Exponential dynamics and crazy topology

Some easy to verify facts:

Have an immediate basin of infinity B


Exponential dynamics and crazy topology

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)


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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)


Exponential dynamics and crazy topology

When , the Julia set

is the unit circle


Exponential dynamics and crazy topology

When , the Julia set

is the unit circle

But when , the

Julia set explodes

A Sierpinskicurve


Exponential dynamics and crazy topology

When , the Julia set

is the unit circle

But when , the

Julia set explodes

B

T

A Sierpinskicurve


Exponential dynamics and crazy topology

When , the Julia set

is the unit circle

But when , the

Julia set explodes

Another Sierpinskicurve


Exponential dynamics and crazy topology

When , the Julia set

is the unit circle

But when , the

Julia set explodes

Also a Sierpinski curve


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

Lots of ways this happens:

parameter plane

when n = 3


Exponential dynamics and crazy topology

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole


Exponential dynamics and crazy topology

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole


Exponential dynamics and crazy topology

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole


Exponential dynamics and crazy topology

Lots of ways this happens:

parameter plane

when n = 3

J is a Sierpinski curve

T

lies in a Sierpinski hole


Exponential dynamics and crazy topology

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,


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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


Exponential dynamics and crazy topology

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.


Exponential dynamics and crazy topology

Corollary:


Exponential dynamics and crazy topology

Corollary:

Yes, those planar topologists

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


Exponential dynamics and crazy topology

Corollary:

Yes, those planar topologists

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

The End!


Exponential dynamics and crazy topology

website:

math.bu.edu/DYSYS


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