Cantor and Sierpinski,

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Cantor and Sierpinski,
Julia and Fatou
Crazy Topology in Complex Dynamics
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Cantor and Sierpinski,
Julia and Fatou
Crazy Topology in Complex Dynamics
Theorem
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Cantor and Sierpinski,
Julia and Fatou
Crazy Topology in Complex Dynamics
Theorem Planar topologists are crazy!
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Cantor and Sierpinski,
Julia and Fatou
Crazy Topology in Complex Dynamics
Theorem Planar topologists are crazy!
Proof Should be clear by the end.....
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Three examples
Cantor bouquets
Indecomposable continua
Sierpinski curves
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Three examples
Cantor bouquets
Indecomposable continua
Sierpinski curves
These arise as Julia sets for
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Example 1 Cantor Bouquets
with Clara Bodelon Michael Hayes Gareth
Roberts Ranjit Bhattacharjee Lee DeVille Monica
Moreno Rocha Kreso Josic Alex Frumosu Eileen Lee
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Orbit of z
Question What is the fate of orbits?
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Julia set of

J closure of orbits that escape to
closure repelling periodic orbits
chaotic set
Fatou set
complement of J
predictable set
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Example 1
is a Cantor bouquet
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Example 1
is a Cantor bouquet
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Example 1
is a Cantor bouquet
attracting fixed point
q
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Example 1
is a Cantor bouquet
q
p
repelling fixed point
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Example 1
is a Cantor bouquet
q
p
x0
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So where is J?
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So where is J?
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So where is J?
Green points lie in the Fatou set
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So where is J?
Green points lie in the Fatou set
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So where is J?
Green points lie in the Fatou set
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So where is J?
Green points lie in the Fatou set
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So where is J?
Green points lie in the Fatou set
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The Julia set is a collection of curves (hairs)
in the right half plane, each with an
endpoint and a stem.
hairs
endpoints
stems
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A Cantor bouquet
q
p
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Colored points escape to and so are in the
Julia set.
q
p
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One such hair lies on the real axis.
repelling fixed point
stem
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Orbits of points on the stems all tend to .
hairs
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So bounded orbits lie in the set of endpoints.
hairs
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So bounded orbits lie in the set of endpoints.
Repelling cycles lie in the set of endpoints.
hairs
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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.
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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.
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S
Some Facts
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S
Some Facts
The only accessible points in J from the Fatou
set are the endpoints you cannot touch the stems
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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 is totally disconnected...
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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 is totally disconnected...
but the endpoints together with the point at
infinity is connected (Mayer)
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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 is totally disconnected...
but the endpoints together with the point at
infinity is connected (Mayer)
Hausdorff dimension of stems 1...
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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 is totally disconnected...
but the endpoints together with the point at
infinity is connected (Mayer)
Hausdorff dimension of stems 1...
but the Hausdorff dimension of endpoints
2! (Karpinska)
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Example 2 Indecomposable Continua
with Nuria Fagella Xavier Jarque Monica Moreno
Rocha
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When
,
undergoes a saddle node bifurcation,
but much more happens...
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As increases through 1/e,
explodes.
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(Sullivan, Goldberg, Keen)
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As
increases through
,
however
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As
increases through
,
however
No new periodic cycles are born
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As
increases through
,
however
No new periodic cycles are born
All move continuously to fill in the plane
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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.
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An indecomposable continuum is a compact,
connected set that cannot be broken into the
union of two (proper) compact, connected subsets.
For example
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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
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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)
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An indecomposable continuum is a compact,
connected set that cannot be broken into the
union of two (proper) compact, connected subsets.
For example indecomposable?
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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
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An indecomposable continuum is a compact,
connected set that cannot be broken into the
union of two (proper) compact, connected subsets.
For example indecomposable?
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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
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Knaster continuum
A well known example of an indecomposable
continuum
Start with the Cantor middle-thirds set
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Knaster continuum
Connect symmetric points about 1/2 with
semicircles
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Knaster continuum
Do the same below about 5/6
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Knaster continuum
And continue....
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Knaster continuum
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Properties of K
There is one curve that passes through all
the endpoints of the Cantor set.
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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.
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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.
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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.
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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....
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Indecomposable!
Try to write K as the union of two compact,
connected sets.
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Indecomposable!
Cant divide it this way.... subsets are
closed but not connected.
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Indecomposable!
Or this way... again closed but not
connected.
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Indecomposable!
Or the union of the outer curve and all the
inaccessible curves ... not closed.
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How the hairs become indecomposable
repelling fixed pt
.
.
.
.
... .
.
.
.
.
attracting fixed pt
stem
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How the hairs become indecomposable
.
.
.
.
... .
.
.
.
.
.
.
.
.
.
2 repelling fixed points
.
.
.
.
.
.
.
Now all points in R escape, so the hair is much
longer
.
.
.
.
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But the hair is even longer!
0
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But the hair is even longer!
0
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But the hair is even longer! And longer.
0
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But the hair is even longer! And longer...
0
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But the hair is even longer! And longer.......
0
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But the hair is even longer! And
longer.............
0
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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
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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 Conject
ure the continuum for each parameter is
topologically distinct.
sin(z)
Sierpinski
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A pair of Cantor bouquets
Julia set of sin(z)
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A pair of Cantor bouquets
Unlike the exponential bouquets (which have
measure 0), these have infinite Lebesgue
measure, though they are homeomorphic to the
exponential bouquets.
Julia set of sin(z)
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sin(z)
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sin(z)
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sin(z)
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(1.2i) sin(z)
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(1 ci) sin(z)
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Questions
Do the hairs become indecomposable continua as in
the exponential case?
If so, what is the topology of these sets?
516
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
517
Sierpinski Curve
A Sierpinski curve is any planar set that is
homeomorphic to the Sierpinski carpet fractal.
The Sierpinski Carpet
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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
519
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....
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can be embedded inside
The topologists sine curve
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can be embedded inside
The topologists sine curve
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can be embedded inside
The topologists sine curve
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
530
can be embedded inside
The Knaster continuum
531
can be embedded inside
The Knaster continuum
532
can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
540
can be embedded inside
The Knaster continuum
541
can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
The Knaster continuum
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can be embedded inside
Even this curve
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Some easy to verify facts
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Some easy to verify facts
Have an immediate basin of infinity B
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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)
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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
562
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)
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When , the Julia set is the unit circle
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But when , the Julia set explodes
When , the Julia set is the unit circle
A Sierpinski curve
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But when , the Julia set explodes
When , the Julia set is the unit circle
B
T
A Sierpinski curve
566
But when , the Julia set explodes
When , the Julia set is the unit circle
Another Sierpinski curve
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But when , the Julia set explodes
When , the Julia set is the unit circle
Also 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.
569
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.
570
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.
571
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.
572
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.
573
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.
574
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.
575
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.
576
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.
577
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.
578
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.
579
Lots of ways this happens
parameter plane when n 3
580
Lots of ways this happens
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
581
Lots of ways this happens
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
582
Lots of ways this happens
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
583
Lots of ways this happens
parameter plane when n 3
J is a Sierpinski curve
T
lies in a Sierpinski hole
584
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,
585
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
586
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
Sorry. I forgot to indicate their locations.
587
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
588
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
589
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
590
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.
591
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
592
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
593
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
594
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
595
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
596
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
597
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.
598
Corollary
599
Corollary
Yes, those planar
topologists are crazy, but I sure wish I were one
of them!
600
Corollary
Yes, those planar
topologists are crazy, but I sure wish I were one
of them!
The End!
601
website
math.bu.edu/DYSYS