1. Cantor and Sierpinski,
Julia and Fatou;
Crazy Topology in Complex Dynamics

6. Cantor and Sierpinski,
Julia and Fatou;
Crazy Topology in Complex Dynamics
Theorem:

7. Cantor and Sierpinski,
Julia and Fatou;
Crazy Topology in Complex Dynamics
Theorem: Planar topologists are crazy!

8. Cantor and Sierpinski,
Julia and Fatou;
Crazy Topology in Complex Dynamics
Theorem: Planar topologists are crazy!
Proof: Should be clear by the end.....

9. Three examples:
Cantor
bouquets
Indecomposable
continua
Sierpinski
curves

10. Three examples:
Cantor
bouquets
Indecomposable
continua
Sierpinski
curves
These arise as Julia sets for:

11. Example 1: Cantor Bouquets
with
Clara Bodelon
Michael Hayes
Gareth Roberts
Ranjit Bhattacharjee
Lee DeVille
Monica Moreno Rocha
Kreso Josic
Alex Frumosu
Eileen Lee

12. Orbit of z:
Question: What is the fate of orbits?

13. Julia set of
J = closure of {orbits that escape to }
= closure {repelling periodic orbits}
= {chaotic set}
Fatou set
= complement of J
= predictable set

14. Example 1:
is a “Cantor bouquet”

15. Example 1:
is a “Cantor bouquet”

16. Example 1:
is a “Cantor bouquet”
attracting
fixed point q

17. Example 1:
is a “Cantor bouquet” q p
repelling
fixed point

18. Example 1:
is a “Cantor bouquet” q x0 p

19. So where is J?

20. So where is J?

21. So where is J?
Green points lie in the Fatou set

22. So where is J?
Green points lie in the Fatou set

23. So where is J?
Green points lie in the Fatou set

24. So where is J?
Green points lie in the Fatou set

25. So where is J?
Green points lie in the Fatou set

26. The Julia set is a collection of curves (hairs) in the right half plane, each with an endpoint
and a stem.
hairs
endpoints
stems

27. A “Cantor bouquet” q p

28. Colored points escape to and so are in the Julia set.q p

33. One such hair lies on the real axis.
repelling
fixed point
stem

34. Orbits of points on the stems all tend to .
hairs

35. So bounded orbits lie in the set of endpoints.
hairs

36. So bounded orbits lie in the set of endpoints.
Repelling cycles lie
in the set of endpoints.
hairs

37. 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.

38. 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.

39. S
Some Facts:

40. The only accessible points in J from the Fatou set are the endpoints;
Some Facts:
The only accessible points in J from
the Fatou set are the endpoints;
you cannot touch the stems

41. The only accessible points in J from the Fatou set are the endpoints;
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...

42. The only accessible points in J from the Fatou set are the endpoints;
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)

43. The only accessible points in J from the Fatou set are the endpoints;
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...

44. The only accessible points in J from the Fatou set are the endpoints;
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)

45. Example 2: Indecomposable Continua
with
Nuria Fagella
Xavier Jarque
Monica Moreno Rocha

46. When ,
undergoes a saddle node bifurcation,
but much more happens...

47. As increases through 1/e, explodes.

272. (Sullivan, Goldberg, Keen)

273. As
increases through ,
; however:

274. As
increases through ,
; however:
No new periodic cycles are born;

275. As
increases through ,
; however:
No new periodic cycles are born;
All move continuously to fill in the plane;

276. 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.”

277. An indecomposable continuum is a compact, connected
set that cannot be broken into the union of two (proper)
compact, connected subsets.
For example:

278. An indecomposable continuum is a compact, connected
set that cannot be broken into the union of two (proper)
compact, connected subsets.
For example: indecomposable? 1

279. 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. 1
(subsets need not be disjoint)

280. An indecomposable continuum is a compact, connected
set that cannot be broken into the union of two (proper)
compact, connected subsets.
For example: indecomposable?

281. 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.

282. An indecomposable continuum is a compact, connected
set that cannot be broken into the union of two (proper)
compact, connected subsets.
For example: indecomposable?

283. 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.

284. Start with the Cantor middle-thirds set
Knaster continuum
A well known example of an indecomposable continuum
Start with the Cantor middle-thirds set

285. Knaster continuum
Connect symmetric points about 1/2 with semicircles

286. Knaster continuum
Do the same below about 5/6

287. Knaster continuum
And continue....

288. Knaster continuum

289. Properties of K: There is one curve that passes through all the
endpoints of the Cantor
set.

290. 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.

291. 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.

292. 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.

293. 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....

294. Indecomposable!
Try to write K as the union
of two compact, connected sets.

295. Indecomposable!
Can’t divide it this way....
subsets are closed
but not connected.

296. Indecomposable!
Or this way...
again closed but
not connected.

297. Indecomposable!
Or the union of the outer curve and all
the inaccessible curves ... not closed.

298. How the hairs become indecomposable:
repelling
fixed pt . . . .
... . . . . .
attracting
fixed pt
stem

299. How the hairs become indecomposable:. . . .
... . . . . . . . . .
2 repelling
fixed points . . . . . . . .
Now all points in R escape,
so the hair is much longer . . . .

300. But the hair is even longer!

301. But the hair is even longer!

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

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

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

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

306. 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.

307. 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)
Sierpinski

308. A pair of Cantor bouquets
Julia set of sin(z)

309. Julia set of sin(z) 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)

310. sin(z)

311. sin(z)

312. sin(z)

313. (1+.2i) sin(z)

314. (1+ ci) sin(z)

515. 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

518. 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. A Sierpinski curve is a universal plane continuum:
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....

520. The topologist’s sine curve
can be embedded inside

521. The topologist’s sine curve
can be embedded inside

522. The topologist’s sine curve
can be embedded inside

523. The Knaster continuum
can be embedded inside

524. The Knaster continuum
can be embedded inside

525. The Knaster continuum
can be embedded inside

526. The Knaster continuum
can be embedded inside

527. The Knaster continuum
can be embedded inside

528. The Knaster continuum
can be embedded inside

529. The Knaster continuum
can be embedded inside

530. The Knaster continuum
can be embedded inside

531. The Knaster continuum
can be embedded inside

532. The Knaster continuum
can be embedded inside

533. The Knaster continuum
can be embedded inside

534. The Knaster continuum
can be embedded inside

535. The Knaster continuum
can be embedded inside

536. The Knaster continuum
can be embedded inside

537. The Knaster continuum
can be embedded inside

538. The Knaster continuum
can be embedded inside

539. The Knaster continuum
can be embedded inside

540. The Knaster continuum
can be embedded inside

541. The Knaster continuum
can be embedded inside

542. The Knaster continuum
can be embedded inside

543. The Knaster continuum
can be embedded inside

544. The Knaster continuum
can be embedded inside

545. The Knaster continuum
can be embedded inside

546. The Knaster continuum
can be embedded inside

547. The Knaster continuum
can be embedded inside

548. The Knaster continuum
can be embedded inside

549. The Knaster continuum
can be embedded inside

550. The Knaster continuum
can be embedded inside

551. The Knaster continuum
can be embedded inside

552. The Knaster continuum
can be embedded inside

553. The Knaster continuum
can be embedded inside

554. The Knaster continuum
can be embedded inside

555. The Knaster continuum
can be embedded inside

556. The Knaster continuum
can be embedded inside

557. Even this “curve”
can be embedded inside

558. Some easy to verify facts:

559. Some easy to verify facts:
Have an immediate basin of infinity B

560. 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)

561. 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)

563. When , the Julia set
is the unit circle

564. When , the Julia set
is the unit circle
But when , the
Julia set explodes
A Sierpinski curve

565. B T When , the Julia set is the unit circle But when , the
Julia set explodes B T
A Sierpinski curve

566. When , the Julia set
is the unit circle
But when , the
Julia set explodes
Another Sierpinski curve

567. Also a Sierpinski curve
When , the Julia set
is the unit circle
But when , the
Julia set explodes
Also a Sierpinski curve

568. 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. lies in a Sierpinski hole
Lots of ways this happens:
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

581. lies in a Sierpinski hole
Lots of ways this happens:
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

582. lies in a Sierpinski hole
Lots of ways this happens:
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

583. lies in a Sierpinski hole
Lots of ways this happens:
parameter plane
when n = 3
J is a Sierpinski curve T
lies in a Sierpinski hole

584. 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,

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: ,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. Black regions are the basin
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