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

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

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

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

Three examples: Cantor bouquets Indecomposable continua Sierpinski curves

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

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

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

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

Example 1: is a “Cantor bouquet”

Example 1: is a “Cantor bouquet”

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

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

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

So where is J?

So where is J?

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

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

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

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

So where is J? 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” q p

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

One such hair lies on the real axis. repelling fixed point 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. hairs So the endpoints are dense in the bouquet.

So bounded orbits lie in the set of endpoints. Repelling cycles lie in the set of endpoints. hairs So the endpoints are dense in the bouquet.

S Some Facts:

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

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...

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)

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...

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)

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

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

As increases through 1/e, explodes.

(Sullivan, Goldberg, Keen)

As increases through , ; however:

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

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

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.”

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? 1

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)

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.

Start with the Cantor middle-thirds set 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: repelling fixed pt . . . . ... . . . . . attracting fixed pt stem

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

But the hair is even longer!

But the hair is even longer!

But the hair is even longer! And longer.

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

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

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

Compactify to get a single curve in a compact region in the plane that accumulates everywhere on itself. The closure is then an indecomposable continuum.

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

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

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)

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?

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

Sierpinski Curve A Sierpinski curve is any planar set that is homeomorphic to the Sierpinski carpet fractal. 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 is a Sierpinski curve. The Sierpinski Carpet

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....

The topologist’s sine curve can be embedded inside

The topologist’s sine curve can be embedded inside

The topologist’s sine curve can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

The Knaster continuum can be embedded inside

Even this “curve” can be embedded inside

Some easy to verify facts:

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

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)

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

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)

When , the Julia set is the unit circle

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

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

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

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

Sierpinski curves arise in lots of different ways in these families: 1. If the critical orbits eventually fall into the trap door (which is disjoint from B), then J is a Sierpinski curve.

Sierpinski curves arise in lots of different ways in these families: 1. If the critical orbits eventually fall into the trap door (which is disjoint from B), then J is a Sierpinski curve.

Sierpinski curves arise in lots of different ways in these families: 1. If the critical orbits eventually fall into the trap door (which is disjoint from B), then J is a Sierpinski curve.

Sierpinski curves arise in lots of different ways in these families: 1. If the critical orbits eventually fall into the trap door (which is disjoint from B), then J is a Sierpinski curve.

Sierpinski curves arise in lots of different ways in these families: 1. If the critical orbits eventually fall into the trap door (which is disjoint from B), then J is a Sierpinski curve.

Sierpinski curves arise in lots of different ways in these families: 1. If the critical orbits eventually fall into the trap door (which is disjoint from B), then J is a Sierpinski curve.

Sierpinski curves arise in lots of different ways in these families: 1. If the critical orbits eventually fall into the trap door (which is disjoint from B), then J is a Sierpinski curve.

Sierpinski curves arise in lots of different ways in these families: 1. If the critical orbits eventually fall into the trap door (which is disjoint from B), then J is a Sierpinski curve.

Sierpinski curves arise in lots of different ways in these families: 1. If the critical orbits eventually fall into the trap door (which is disjoint from B), then J is a Sierpinski curve.

Sierpinski curves arise in lots of different ways in these families: 1. If the critical orbits eventually fall into the trap door (which is disjoint from B), then J is a Sierpinski curve.

Lots of ways this happens: parameter plane when n = 3

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

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

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

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

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,

Theorem: Two maps drawn from the same Sierpinski hole have the same dynamics, but those drawn from different holes are not conjugate (except in very few symmetric cases). n = 4, escape time 4, 24 Sierpinski holes, but only five conjugacy classes

Theorem: Two maps drawn from the same Sierpinski hole have the same dynamics, but those drawn from different holes are not conjugate (except in very few symmetric cases). n = 4, escape time 12: 402,653,184 Sierpinski holes, but only 67,108,832 distinct conjugacy classes Sorry. I forgot to indicate their locations.

Sierpinski curves arise in lots of different ways in these families: 2. If the parameter lies in the main cardioid of a buried baby Mandelbrot set, J is again a Sierpinski curve. parameter plane when n = 4

Sierpinski curves arise in lots of different ways in these families: 2. If the parameter lies in the main cardioid of a buried baby Mandelbrot set, J is again a Sierpinski curve. parameter plane when n = 4

Sierpinski curves arise in lots of different ways in these families: 2. If the parameter lies in the main cardioid of a buried baby Mandelbrot set, J is again a Sierpinski curve. parameter plane when n = 4

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.

Sierpinski curves arise in lots of different ways in these families: 3. If the parameter lies at a buried point in the “Cantor necklaces” in the parameter plane, J is again a Sierpinski curve. parameter plane n = 4

Sierpinski curves arise in lots of different ways in these families: 3. If the parameter lies at a buried point in the “Cantor necklaces” in the parameter plane, J is again a Sierpinski curve. parameter plane n = 4

Sierpinski curves arise in lots of different ways in these families: 3. If the parameter lies at a buried point in the “Cantor necklaces” in the parameter plane, J is again a Sierpinski curve. parameter plane n = 4

Sierpinski curves arise in lots of different ways in these families: 4. There is a Cantor set of circles in the parameter plane on which each parameter corresponds to a Sierpinski curve. n = 3

Sierpinski curves arise in lots of different ways in these families: 4. There is a Cantor set of circles in the parameter plane on which each parameter corresponds to a Sierpinski curve. n = 3

Sierpinski curves arise in lots of different ways in these families: 4. There is a Cantor set of circles in the parameter plane on which each parameter corresponds to a Sierpinski curve. n = 3

Theorem: All these Julia sets are the same topologically, but they are all (except for symmetrically located parameters) VERY different from a dynamics point of view (i.e., the maps are not conjugate). Problem: Classify the dynamics on these Sierpinski curve Julia sets.

Corollary:

Corollary: Yes, those planar topologists are crazy, but I sure wish I were one of them!

Corollary: Yes, those planar topologists are crazy, but I sure wish I were one of them! The End!

website: math.bu.edu/DYSYS