Exponential Dynamics and (Crazy) Topology Cantor bouquetsIndecomposable continua.

Exponential Dynamics and (Crazy) TopologyCantor bouquetsIndecomposable continua

Cantor bouquetsIndecomposable continuaThese are Julia sets ofExponential Dynamics and (Crazy) Topology

Example 1: Cantor BouquetswithClara BodelonMichael HayesGareth RobertsRanjit BhattacharjeeLee DeVilleMonica Moreno RochaKreso JosicAlex FrumosuEileen 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 thecritical points that determined everything.But has no critical points.

For polynomials, it was the orbit of thecritical points that determined everything.But has no critical points.But 0 is an asymptotic value; any farleft half-plane is wrapped infinitely oftenaround 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 bouquetattracting fixed pointq

Example 1: is a Cantor bouquetattracting fixed pointqThe orbit of 0 alwaystends this attractingfixed point

Example 1: is a Cantor bouquetqprepelling fixed point

Example 1: is a Cantor bouquetqpx0

Example 1: is a Cantor bouquetqpx0And for all

So where is J?

So where is J?in this half plane

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

hairsstemsendpointsThe Julia set is a collection of curves (hairs) in the right half plane, each with an endpointand a stem.

A Cantor bouquetpq

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

One such hair lies on the real axis.stemrepellingfixed point

hairsOrbits of points on the stems all tend to .

hairsSo bounded orbits lie in the set of endpoints.

hairsSo bounded orbits lie in the set of endpoints.Repelling cycles lie in the set of endpoints.

hairsSo bounded orbits lie in the set of endpoints.Repelling cycles lie in the set of endpoints.So the endpoints aredense in the bouquet.

SSome Facts:

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

SSome Crazy Facts:The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stemsThe set of endpoints together with thepoint at infinity is connected ...

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

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

SSome Crazy Facts:The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stemsThe set of endpoints together with thepoint 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 pairof Cantor bouquets

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

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

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

Another example:The lines y = c are bothwrapped around an ellipsewith foci at c-c

Another example:The lines y = c are bothwrapped around an ellipsewith foci at , and all orbitsin the ellipse tend to 0 ifc is small enoughc-c

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

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

Another example:

c-cThe vertical lines givenby x = n + /2 arealso in the Fatou set.

And all points in the preimages of the striplie in the Fatou set...

And so on to get anotherCantor bouquet.

The difference here is that the Cantor bouquet forthe sine function has infinite Lebesgue measure,while the exponential bouquet has zero measure.

Example 2: Indecomposable ContinuawithNuria FagellaXavier JarqueMonica Moreno Rocha

When, undergoes a saddle node bifurcation,The two fixed points coalesceand 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 theJulia set is the entire complex plane.

Asincreases through,; however:

Asincreases throughNo new periodic cycles are born;,; however:

Asincreases throughNo new periodic cycles are born;,; however:All move continuously to fill in the plane densely;

Asincreases throughNo 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, connectedset that cannot be broken into the union of two (proper)compact, connected subsets.For example:

An indecomposable continuum is a compact, connectedset that cannot be broken into the union of two (proper)compact, connected subsets.For example: indecomposable?01

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

An indecomposable continuum is a compact, connectedset that cannot be broken into the union of two (proper)compact, connected subsets.For example: No, decomposable.

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

An indecomposable continuum is a compact, connectedset that cannot be broken into the union of two (proper)compact, connected subsets.For example: No, decomposable.

Knaster continuumStart with the Cantor middle-thirds setA well known example of an indecomposable continuum

Knaster continuumConnect symmetric points about 1/2 with semicircles

Knaster continuumDo the same below about 5/6

Knaster continuumAnd continue....

Knaster continuum

Properties of K:There is one curve thatpasses through all theendpoints of the Cantorset.

There is one curve thatpasses through all theendpoints of the Cantorset.It accumulates everywhere onitself and on K.Properties of K:

There is one curve thatpasses through all theendpoints of the Cantorset.It accumulates everywhere onitself and on K.And is the only pieceof K that is accessiblefrom the outside.Properties of K:

There is one curve thatpasses through all theendpoints of the Cantorset.It accumulates everywhere onitself and on K.And is the only pieceof K that is accessiblefrom 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 thatpasses through all theendpoints of the Cantorset.It accumulates everywhere onitself and on K.And is the only pieceof K that is accessiblefrom 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 unionof two compact, connected sets.

Indecomposable!Cant 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:........... .attractingfixed ptrepelling fixed ptstem

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

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

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

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 pointall other orbits either tend toor 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 bouquetsA 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 indecomposablecontinua as in the exponential case?If so, what is the topology of these sets?

Parameter plane for To plot the parameter plane (the analogueof the Mandelbrot set), for each plot thecorresponding orbit of 0.

If 0 escapes, the color ; J is the entire plane.

If 0 does not escape, leave black; J isusually a pinched Cantor bouquet.

Parameter plane for

has an attracting fixed point in thiscardioid1Parameter plane for

Period 2 region12Parameter plane for

12Parameter plane for 334455

Period 2 regionParameter plane for 12So undergoesa period doublingbifurcation alongthis path in theparameter planeat Fixed point bifurcations

The Cantor bouquet

The Cantor bouquetA repelling 2-cycle attwo endpoints

The hairs containing the 2-cycle meet at theneutral fixed point, and then remain attached.Meanwhile an attracting 2 cycle emerges.We get a pinched Cantor bouquet

Period 3 regionOther bifurcationsParameter plane for 123

Period tripling bifurcationOther bifurcationsParameter plane for 123

Three hairs containing a 3-cycle meet at theneutral fixed point, and then remain attachedWe get a different pinched Cantor bouquet

Period 5bifurcationOther bifurcationsParameter plane for 1235

Five hairs containing a 5-cycle meet at theneutral fixed point, and then remain attachedWe get a different pinched Cantor bouquet

Lots of explosions occur....13

Lots of explosions occur....

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 BlanchardToni GarijoMatt HolzerU. HoomiforgotDan LookSebastian Marotta Mark MorabitoMonica Moreno RochaKevin PilgrimElizabeth RussellYakov ShapiroDavid Uminsky

with:Example 3: Sierpinski Curves

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

The Sierpinski CarpetTopological CharacterizationAny 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 topologists sine curvecan be embedded inside

The Knaster continuumcan be embedded inside

can be embedded insideEven this curve

Some easy to verify facts:

Have an immediate basin of infinity BSome easy to verify facts:

Have an immediate basin of infinity B0 is a pole so have a trap door T (the preimage of B)Some easy to verify facts:

Have an immediate basin of infinity B0 is a pole so have a trap door T (the preimage of B)2n critical points given by but really onlyone critical orbit due to symmetrySome easy to verify facts:

J is now the boundary of the escaping orbits (not the closure)Have an immediate basin of infinity B0 is a pole so have a trap door T (the preimage of B)2n critical points given by but really onlyone critical orbit due to symmetrySome easy to verify facts:

When , the Julia set is the unit circle

But when , theJulia set explodesA Sierpinski curveWhen , the Julia set is the unit circle

But when , theJulia set explodesA Sierpinski curveWhen , the Julia set is the unit circleBT

But when , theJulia set explodesWhen , the Julia set is the unit circleAnother Sierpinski curve

But when , theJulia set explodesWhen , the Julia set is the unit circleAlso a Sierpinski curve

Sierpinski curves arise in lots of different ways in these families:1. If the critical orbitseventually fall intothe trap door (whichis disjoint from B),then J is a Sierpinskicurve.

Lots of ways this happens:parameter planewhen n = 3

Lots of ways this happens:Tparameter planewhen n = 3J is a Sierpinski curvelies in a Sierpinski hole

Theorem: Two maps drawn from the same Sierpinskihole have the same dynamics, but those drawn fromdifferent holes are not conjugate (except in veryfew symmetric cases).

n = 4, escape time 4, 24 Sierpinski holes,


n = 4, escape time 4, 24 Sierpinski holes,but only five conjugacy classes


n = 4, escape time 12: 402,653,184 Sierpinski holes,but only 67,108,832 distinct conjugacy classesSorry. 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.Black regions are the basin of an attracting cycle.

Sierpinski curves arise in lots of different ways in these families:3. If the parameter liesat a buried point in theCantor necklaces inthe parameter plane,J is again a Sierpinskicurve.parameter plane n = 4

Sierpinski curves arise in lots of different ways in these families:4. There is a Cantor setof circles in the parameterplane on which eachparameter correspondsto 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 theseSierpinski curve Julia sets.

Corollary:

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

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

The End!

math.bu.edu/DYSYSwebsite:

Exponential Dynamics and (Crazy) Topology Cantor bouquetsIndecomposable continua.

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set of endpoints

chaotic set

predictable set

green points

endpoints aredense

accessible points

cantor bouquetpqcolored

complement of j