Exponential Dynamics and (Crazy) Topology
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Cantor bouquets
Indecomposable continua
Exponential Dynamics and (Crazy) Topology
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Cantor bouquets
Indecomposable continua
These are Julia sets of
 exp( z), 0   1/e
 exp( z),  1/e
Example 1: Cantor Bouquets
E (z)   exp( z)


0   1/e
with
Clara Bodelon
 Hayes
Michael
Gareth Roberts
Ranjit Bhattacharjee
Lee DeVille
Monica Moreno Rocha
Kreso Josic
Alex Frumosu
Eileen Lee
E (z)   exp( z)
Orbit of z:

e z
z
z, e , e ,...

z, E (z), E (E (z))  E2 (z), E3 (z),...


 

Question: What is the fate of orbits?


Julia set of E  J(E )
J = closure of {orbits that escape to
} *
 {repelling periodic orbits}
= closure
= {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 E  has no critical points.

For polynomials, it was the orbit of the
critical points that determined everything.
But E  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:
E (z)   exp( z)
0   1/e
J(E ) is a “Cantor bouquet”




Example 1:
E (z)   exp( z)
0   1/e
J(E ) is a “Cantor bouquet”



3
yx
y  e x

2


1
-1
1
-1
2
3
Example 1:
E (z)   exp( z)
0   1/e
J(E ) is a “Cantor bouquet”



3
yx
y  e x

2
attracting
fixed point


1
q
-1
-1
1
2
3
Example 1:
E (z)   exp( z)
0   1/e
J(E ) is a “Cantor bouquet”



3
yx
y  e x

2
attracting
fixed point


The orbit of 0 always
tends this attracting
fixed point
1
q
-1
-1
1
2
3
Example 1:
E (z)   exp( z)
0   1/e
J(E ) is a “Cantor bouquet”



3
yx
y  e x

2


1
q
-1
1
p2
repelling
fixed point
-1
3
Example 1:
E (z)   exp( z)
0   1/e
J(E ) is a “Cantor bouquet”



3
yx
y  e x

2


E '(x0 )  1  E (x0 )  x0
1
q
-1

-1
1
x0
p2
 
3
Example 1:
E (z)   exp( z)
0   1/e
J(E ) is a “Cantor bouquet”



3
yx
y  e x

2
And E '(x) 1
for all x  x0




E '(x0 )  1  E (x0 )  x0
1
q
-1

-1
1
x0
p2
 
3
So where is J?
3


q





3
x  x0
So where is J?
3


q





3
x  x0
So where is J?
3
| E '(z) |  1
in this half plane


q





3
x  x0
So where is J?
3



x  x0



3
Green points lie in the Fatou set
So where is J?
3



x  x0



3
Green points lie in the Fatou set
So where is J?
3





3
Green points lie in the Fatou set
So where is J?
3





3
Green points lie in the Fatou set
So where is J?
3





3
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”
Quic kTime™ and a
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q
p
Quic kTime™ and a
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Colored points escape to 
and so now are in the Julia set.

Quic kTime™ and a
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q
p
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One such hair lies on the real axis.
3
2
1
-1
1
-1
2
repelling
fixed point
3
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.
So the endpoints are
dense in the bouquet.
hairs
So bounded orbits lie in the set of endpoints.
Repelling cycles lie
in the set of endpoints.
So the endpoints are
dense in the bouquet.
hairs
Some Facts:
Quic kTime™ and a
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S
Quic kTime™ and a
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Quic kTime™ and a
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Quic kTime™ and a
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Some Facts:
Quic kTime™ and a
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The only accessible points in J from
the Fatou set are the endpoints;
you cannot touch the stems
S
Quic kTime™ and a
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Quic kTime™ and a
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Quic kTime™ and a
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Some Crazy Facts:
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The only accessible points in J from
the Fatou set are the endpoints;
you cannot touch the stems
S
The set of endpoints together with the
point at infinity is connected ...
Quic kTime™ and a
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Quic kTime™ and a
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Some Crazy Facts:
Quic kTime™ and a
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The only accessible points in J from
the Fatou set are the endpoints;
you cannot touch the stems
S
The set of endpoints together with the
point at infinity is connected ...
but the set of endpoints is
totally disconnected (Mayer)
Quic kTime™ and a
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Quic kTime™ and a
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Quic kTime™ and a
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Some Crazy Facts:
Quic kTime™ and a
TIFF ( LZ W ) dec ompres s or
are needed to s ee t his pic tur e.
The only accessible points in J from
the Fatou set are the endpoints;
you cannot touch the stems
S
The set of endpoints together with the
point at infinity is connected ...
but the set of endpoints is
totally disconnected (Mayer)
Quic kTime™ and a
TIFF ( LZ W ) dec ompres s or
are needed to s ee t his pic tur e.
Quic kTime™ and a
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Hausdorff dimension of {stems} = 1...
Quic kTime™ and a
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Some Crazy Facts:
Quic kTime™ and a
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The only accessible points in J from
the Fatou set are the endpoints;
you cannot touch the stems
S
The set of endpoints together with the
point at infinity is connected ...
but the set of endpoints is
totally disconnected (Mayer)
Quic kTime™ and a
TIFF ( LZ W ) dec ompres s or
are needed to s ee t his pic tur e.
Quic kTime™ and a
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Hausdorff dimension of {stems} = 1...
but the Hausdorff dimension of
{endpoints} = 2! (Karpinska)
Quic kTime™ and a
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Another example:  sin(z), 0  1

Looks a little different, but still a pair
of Cantor bouquets
Another example:  sin(z), 0  1





The interval [-, ] is
contracted inside itself,
and all these orbits tend to
0 (so are in the Fatou set)
Another example:  sin(z), 0  1





The real line is contracted
onto the interval [ ,  ] ,
and all these orbits tend to 0
(so are in the Fatou set)
Another example:  sin(z), 0  1

-/2
/2
The vertical lines x = n + /2 are
mapped to either [, ∞) or (-∞, - ],
so these lines are in the Fatou set....
Another example:  sin(z), 0  1

c



 -c
The lines y =  c are both
wrapped around an ellipse
with foci at 

Another example:  sin(z), 0  1

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:  sin(z), 0  1

c
-c
So all points in the ellipse
lie in the Fatou set
Another example:  sin(z), 0  1

c
-c
So do all points in the
strip | y |  c
Another example:  sin(z), 0  1

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.
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Example 2: Indecomposable Continua
E (z)   exp( z)


 1/e
with
Nuria Fagella
Jarque
Xavier
Monica Moreno Rocha
When

 1/e , E
undergoes a “saddle node” bifurcation,
 1/e

 1/e


The two fixed points coalesce
and disappear from the real
axis when  goes above 1/e.
E (z)   exp( z)


 1/e

And now the orbit of 0 goes off to ∞ ....
And as

 increases through 1/e, J(E ) explodes.

E (z)   exp( z)


 1/e

Theorem: If the orbit of 0 goes to ∞, then the
Julia set is the entire complex plane.
(Sullivan, Goldberg, Keen)
As

 increases through 1/e, J(E )  C ; however:


As
 increases through 1/e, J(E )  C ; however:
No new periodic cycles are born;



As
 increases through 1/e, J(E )  C ; however:
No new periodic cycles are born;



All move continuously to fill in the plane densely;
As
 increases through 1/e, J(E )  C ; 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.
For example:
0
No, decomposable.
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.
For example:
No, decomposable.
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.
For example:
No, decomposable.
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:
 1/e
.
..
..
... .
.
..
attracting
fixed pt
repelling
fixed pt
stem
How the hairs become indecomposable:
 1/e
.
..
..
... .
.
..
2 repelling . . .
fixed points ..
..
..
y0
.. .
..
. .
Now all points in R escape,
so the hair is much longer
 1/e
But the hair is even longer!
y0
0
 1/e
But the hair is even longer!
y
E
y0

0
 1/e
But the hair is even longer! And longer.
y
E

y0
0
 1/e
But the hair is even longer! And longer...
y
E

y0
0
 1/e
But the hair is even longer! And longer.......
y
E

y0
0
 1/e
But the hair is even longer! And longer.............
y
E

y0
0
 1/e
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.
y
y0
0
 1/e
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 E
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 E

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Parameter plane for E

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Parameter plane for E

Parameter plane for E

E has an
attracting fixed
point in this
cardioid
1
Parameter plane for E

Period 2 region
2
1
Parameter plane for E
4

5
2
3
1
5
4
3
Fixed point bifurcations
Period 2 region
2
1
So E undergoes
a period doubling
bifurcation along
this path in the
parameter plane
at    e
Parameter plane for E
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
3
2
1
Parameter plane for E
Period 3 region
Other bifurcations
3
2
1
Parameter plane for E
Period tripling
bifurcation
Three hairs containing a 3-cycle meet at the
neutral fixed point, and then remain attached
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We get a different “pinched” Cantor bouquet
Other bifurcations
5
2
3
1
Parameter plane for E
Period 5
bifurcation
Five hairs containing a 5-cycle meet at the
neutral fixed point, and then remain attached
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We get a different “pinched” Cantor bouquet
Lots of explosions occur....
3
1
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Lots of explosions occur....
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4
1
5
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On a path like this, we pass through infinitely
many regions where there is an attracting cycle,
so J is a pinched Cantor bouquet.....
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and infinitely many hairs where J is the entire plane.
slower
Example 3: Sierpinski Curves
F (z)  z 
n

z
 C n  2
n
with:

Paul Blanchard
Garijo
Toni
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.
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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
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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....
The topologist’s sine curve
can be embedded
inside
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The topologist’s sine curve
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The Knaster continuum
can be embedded
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The Knaster continuum
can be embedded
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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The Knaster continuum
can be embedded
inside
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Even this “curve”
can be embedded
inside
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F (z)  z 
n


z
n
  C, n  2
Some easy to verify facts:

F (z)  z 
n


z
n
  C, n  2
Some easy to verify facts:

Have an immediate basin of infinity B
F (z)  z 
n


z
n
  C, n  2
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)
F (z)  z 
n


z
n
  C, n  2
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
1/2n

F (z)  z 
n


z
n
  C, n  2
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
1/2n
J is now the boundary of the escaping orbits
(not the closure) 
F (z)  z 
n
When   0 , the Julia set
is the unit circle


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0


z
n
  C, n  2
F (z)  z 
n
When   0 , the Julia set
is the unit circle




z
n
  C, n  2
But when   0 , the
Julia set explodes

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0
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  1/16, n  2
A Sierpinski curve
F (z)  z 
n
When   0 , the Julia set
is the unit circle




z
n
  C, n  2
But when   0 , the
Julia set explodes
B
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0
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T
  1/16, n  2
A Sierpinski curve
F (z)  z 
n
When   0 , the Julia set
is the unit circle




z
n
  C, n  2
But when   0 , the
Julia set explodes

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0
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  .001, n  2
Another Sierpinski curve
F (z)  z 
n
When   0 , the Julia set
is the unit circle




z
n
  C, n  2
But when   0 , the
Julia set explodes

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0
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  .02, n  2
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.
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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.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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.
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are needed to see this picture.
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.
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Lots of ways this happens:
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parameter plane
when n = 3
Lots of ways this happens:

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole
Lots of ways this happens:

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole
Lots of ways this happens:

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 lies in a Sierpinski hole
Lots of ways this happens:

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
parameter plane
when n = 3
F (v )  T
k

J is a Sierpinski curve
 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,
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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
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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 67,108,832 distinct conjugacy classes
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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:
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Corollary: Yes, those planar topologists
are crazy, but I sure wish I were one of them!
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Corollary: Yes, those planar topologists
are crazy, but I sure wish I were one of them!
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The End!
website:
math.bu.edu/DYSYS