Exponential Dynamics and (Crazy) Topology Cantor bouquetsIndecomposable continua.
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Transcript of Exponential Dynamics and (Crazy) Topology Cantor bouquetsIndecomposable continua.
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Exponential Dynamics and (Crazy) TopologyCantor bouquetsIndecomposable continua
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Cantor bouquetsIndecomposable continuaThese are Julia sets ofExponential Dynamics and (Crazy) Topology
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Example 1: Cantor BouquetswithClara BodelonMichael HayesGareth RobertsRanjit BhattacharjeeLee DeVilleMonica Moreno RochaKreso JosicAlex FrumosuEileen 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** not the boundary of {orbits that escape to }
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For polynomials, it was the orbit of thecritical points that determined everything.But has no critical points.
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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).
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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 bouquetattracting fixed pointq
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Example 1: is a Cantor bouquetattracting fixed pointqThe orbit of 0 alwaystends this attractingfixed point
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Example 1: is a Cantor bouquetqprepelling fixed point
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Example 1: is a Cantor bouquetqpx0
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Example 1: is a Cantor bouquetqpx0And for all
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So where is J?
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So where is J?
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So where is J?in this half plane
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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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hairsstemsendpointsThe Julia set is a collection of curves (hairs) in the right half plane, each with an endpointand a stem.
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A Cantor bouquetpq
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Colored points escape to and so now are in the Julia set.pq
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One such hair lies on the real axis.stemrepellingfixed point
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hairsOrbits of points on the stems all tend to .
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hairsSo bounded orbits lie in the set of endpoints.
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hairsSo bounded orbits lie in the set of endpoints.Repelling cycles lie in the set of endpoints.
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hairsSo bounded orbits lie in the set of endpoints.Repelling cycles lie in the set of endpoints.So the endpoints aredense in the bouquet.
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hairsSo bounded orbits lie in the set of endpoints.Repelling cycles lie in the set of endpoints.So the endpoints aredense in the bouquet.
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SSome Facts:
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SSome Facts:The only accessible points in J from the Fatou set are the endpoints; you cannot touch the stems
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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 ...
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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)
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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...
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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)
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Another example:Looks a little different, but still a pairof Cantor bouquets
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Another example:The interval [-, ] iscontracted inside itself,and all these orbits tend to 0 (so are in the Fatou set)
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Another example:The real line is contractedonto the interval ,and all these orbits tend to 0(so are in the Fatou set)
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Another example:The vertical lines x = n + /2 aremapped to either [, ) or (-, - ],so these lines are in the Fatou set....-/2 /2
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Another example:The lines y = c are bothwrapped around an ellipsewith foci at c-c
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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
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Another example:So all points in the ellipselie in the Fatou set c-c
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Another example:So do all points in the strip c-c
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Another example:
c-cThe vertical lines givenby x = n + /2 arealso in the Fatou set.
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And all points in the preimages of the striplie in the Fatou set...
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And so on to get anotherCantor bouquet.
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The difference here is that the Cantor bouquet forthe sine function has infinite Lebesgue measure,while the exponential bouquet has zero measure.
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Example 2: Indecomposable ContinuawithNuria FagellaXavier JarqueMonica Moreno Rocha
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When, undergoes a saddle node bifurcation,The two fixed points coalesceand disappear from the real axis when goes above 1/e.
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And now the orbit of 0 goes off to ....
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And as increases through 1/e, explodes.
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(Sullivan, Goldberg, Keen)Theorem: If the orbit of 0 goes to , then theJulia set is the entire complex plane.
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Asincreases through,; however:
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Asincreases throughNo new periodic cycles are born;,; however:
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Asincreases throughNo new periodic cycles are born;,; however:All move continuously to fill in the plane densely;
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Asincreases throughNo new periodic cycles are born;,; however: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, connectedset 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, connectedset that cannot be broken into the union of two (proper)compact, connected subsets.For example: indecomposable?01
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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
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An indecomposable continuum is a compact, connectedset 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, connectedset that cannot be broken into the union of two (proper)compact, connected subsets.For example: No, decomposable.
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An indecomposable continuum is a compact, connectedset 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, connectedset that cannot be broken into the union of two (proper)compact, connected subsets.For example: No, decomposable.
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Knaster continuumStart with the Cantor middle-thirds setA well known example of an indecomposable continuum
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Knaster continuumConnect symmetric points about 1/2 with semicircles
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Knaster continuumDo the same below about 5/6
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Knaster continuumAnd continue....
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Knaster continuum
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Properties of K:There is one curve thatpasses through all theendpoints of the Cantorset.
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There is one curve thatpasses through all theendpoints of the Cantorset.It accumulates everywhere onitself and on K.Properties of K:
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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:
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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:
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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:
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Indecomposable! Try to write K as the unionof two compact, connected sets.
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Indecomposable!Cant divide it this way.... subsets are closed but not connected.
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Or this way... again closed but not connected.Indecomposable!
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Or the union of the outer curve and all the inaccessible curves ... not closed.Indecomposable!
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How the hairs become indecomposable:........... .attractingfixed ptrepelling fixed ptstem
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How the hairs become indecomposable:........... .................2 repelling fixed pointsNow 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 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)
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Julia set of sin(z)A pair of Cantor bouquets
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Julia set of sin(z)A pair of Cantor bouquetsA similar explosion occurs for the sine family (1 + ci) 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 indecomposablecontinua as in the exponential case?If so, what is the topology of these sets?
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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.
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Parameter plane for
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Parameter plane for
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Parameter plane for
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has an attracting fixed point in thiscardioid1Parameter plane for
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Period 2 region12Parameter plane for
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12Parameter plane for 334455
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Period 2 regionParameter plane for 12So undergoesa period doublingbifurcation alongthis path in theparameter planeat Fixed point bifurcations
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The Cantor bouquet
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The Cantor bouquetA repelling 2-cycle attwo endpoints
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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
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Period 3 regionOther bifurcationsParameter plane for 123
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Period tripling bifurcationOther bifurcationsParameter plane for 123
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Three hairs containing a 3-cycle meet at theneutral fixed point, and then remain attachedWe get a different pinched Cantor bouquet
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Period 5bifurcationOther bifurcationsParameter plane for 1235
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Five hairs containing a 5-cycle meet at theneutral fixed point, and then remain attachedWe get a different pinched Cantor bouquet
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Lots of explosions occur....13
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Lots of explosions occur....
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145
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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.
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slower
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Paul BlanchardToni GarijoMatt HolzerU. HoomiforgotDan LookSebastian Marotta Mark MorabitoMonica Moreno RochaKevin PilgrimElizabeth RussellYakov ShapiroDavid Uminsky
with:Example 3: Sierpinski Curves
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A Sierpinski curve is any planar set that is homeomorphic to the Sierpinski carpet fractal.The Sierpinski CarpetSierpinski Curve
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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.
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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....
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The topologists sine curvecan be embedded inside
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The topologists sine curvecan be embedded inside
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The topologists sine curvecan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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The Knaster continuumcan be embedded inside
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can be embedded insideEven this curve
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Some easy to verify facts:
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Have an immediate basin of infinity BSome easy to verify facts:
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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:
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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:
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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:
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When , the Julia set is the unit circle
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But when , theJulia set explodesA Sierpinski curveWhen , the Julia set is the unit circle
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But when , theJulia set explodesA Sierpinski curveWhen , the Julia set is the unit circleBT
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But when , theJulia set explodesWhen , the Julia set is the unit circleAnother Sierpinski curve
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But when , theJulia set explodesWhen , the Julia set is the unit circleAlso a Sierpinski curve
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Lots of ways this happens:parameter planewhen n = 3
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Lots of ways this happens:Tparameter planewhen n = 3J is a Sierpinski curvelies in a Sierpinski hole
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Lots of ways this happens:Tparameter planewhen n = 3J is a Sierpinski curvelies in a Sierpinski hole
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Lots of ways this happens:Tparameter planewhen n = 3J is a Sierpinski curvelies in a Sierpinski hole
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Lots of ways this happens:Tparameter planewhen n = 3J is a Sierpinski curvelies in a Sierpinski hole
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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,
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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,but only five conjugacy classes
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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 12: 402,653,184 Sierpinski holes,but only 67,108,832 distinct conjugacy classesSorry. I forgot to indicate their locations.
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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.
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Corollary:
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Corollary: Yes, those planar topologistsare crazy, but I sure wish I were one of them!
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Corollary: Yes, those planar topologistsare crazy, but I sure wish I were one of them!
The End!
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math.bu.edu/DYSYSwebsite: