Exponential Dynamics and (Crazy) Topology Cantor bouquetsIndecomposable continua

Cantor bouquetsIndecomposable continua These are Julia sets of Exponential Dynamics and (Crazy) Topology

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

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 * * not the boundary of {orbits that escape to }

For polynomials, it was the orbit of the critical points that determined everything. But has no critical points.

For polynomials, it was the orbit of the critical points that determined everything. But has no critical points. But 0 is an asymptotic value; any far left half-plane is wrapped infinitely often around 0, just like a critical value. So the orbit of 0 for the exponential plays a similar role as in the quadratic family (only what happens to the Julia sets is very different in this case).

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” attracting fixed point q The orbit of 0 always tends this attracting fixed point

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

Example 1: is a “Cantor bouquet” qpx0x0

Example 1: is a “Cantor bouquet” qpx0x0 And for all

So where is J?

in this half plane

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

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

A “Cantor bouquet” p q

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

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

hairs 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.

S Some Facts:

S The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stems

S Some Crazy Facts: The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stems The set of endpoints together with the point at infinity is connected...

S Some Crazy Facts: The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stems The set of endpoints together with the point at infinity is connected... but the set of endpoints is totally disconnected (Mayer)

S Some Crazy Facts: The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stems The set of endpoints together with the point at infinity is connected... but the set of endpoints is totally disconnected (Mayer) Hausdorff dimension of {stems} = 1...

S Some Crazy Facts: The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stems The set of endpoints together with the point at infinity is connected... but the set of endpoints is totally disconnected (Mayer) Hausdorff dimension of {stems} = 1... but the Hausdorff dimension of {endpoints} = 2! (Karpinska)

Another example: Looks a little different, but still a pair of Cantor bouquets

Another example: The interval [-, ] is contracted inside itself, and all these orbits tend to 0 (so are in the Fatou set)

Another example: The real line is contracted onto the interval, and all these orbits tend to 0 (so are in the Fatou set)

Another example: The vertical lines x = n  +  /2 are mapped to either [, ∞) or (-∞, - ], so these lines are in the Fatou set  /2  /2

Another example: The lines y = c are both wrapped around an ellipse with foci at c -c

Another example: 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 c -c

Another example: So all points in the ellipse lie in the Fatou set c -c

Another example: So do all points in the strip c -c

Another example: 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.

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

When,undergoes a “saddle node” bifurcation, The two fixed points coalesce and disappear from the real axis when goes above 1/e.

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

And as increases through 1/e, explodes.

(Sullivan, Goldberg, Keen) Theorem: If the orbit of 0 goes to ∞, then the Julia set is the entire complex plane.

As increases through,; however:

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

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

As increases through No new periodic cycles are born;,; however: 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? 01

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. (subsets need not be disjoint) 01

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

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.

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.... Properties of K:

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.

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

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

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

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

But the hair is even longer! 0

0

But the hair is even longer! And longer. 0

But the hair is even longer! And longer... 0

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. 0

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)

Julia set of 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)

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 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

has an attracting fixed point in this cardioid 1 Parameter plane for

Period 2 region1 2 Parameter plane for

Period 2 region Parameter plane for 1 2 So undergoes a period doubling bifurcation along this path in the parameter plane at Fixed point bifurcations

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

Period 3 region Other bifurcations Parameter plane for 1 2 3

Period tripling bifurcation Other bifurcations Parameter plane for 1 2 3

Three hairs containing a 3-cycle meet at the neutral fixed point, and then remain attached We get a different “pinched” Cantor bouquet

Period 5 bifurcation Other bifurcations Parameter plane for

Five hairs containing a 5-cycle meet at the neutral fixed point, and then remain attached We get a different “pinched” Cantor bouquet

Lots of explosions occur

1 4 5

On a path like this, we pass through infinitely many regions where there is an attracting cycle, so J is a pinched Cantor bouquet..... and infinitely many hairs where J is the entire plane.

slower

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 with: Example 3: Sierpinski Curves

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

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.

Any planar, one-dimensional, compact, connected set can be homeomorphically embedded in a Sierpinski curve. More importantly.... A Sierpinski curve is a universal plane continuum: 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”

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:

J is now the boundary of the escaping orbits (not the closure) 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:

When, the Julia set is the unit circle

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

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

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

But when, the Julia set explodes When, the Julia set is the unit circle 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

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

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

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

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

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

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:

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!

math.bu.edu/DYSYS website: