Dynamics of the family of complex maps Paul Blanchard Toni Garijo Matt Holzer Robert Kozma Dan Look Sebastian Marotta Mark Morabito with: Monica Moreno Rocha Kevin Pilgrim Elizabeth Russell Yakov Shapiro David Uminsky Sum Wun Ellce Sierpinski Galore

Sierpinski curves show up as Julia sets for many different parameters

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

Much of what we’ll see holds for the more general family where c is the center of a hyperbolic component of the Multibrot set

where c is the center of a hyperbolic component of the Multibrot set But for simplicity, we’ll concentrate for the most part on the easier family Much of what we’ll see holds for the more general family

Why the interest in these maps?

1. These are singular perturbations of z n.

Why the interest in these maps? 1. These are singular perturbations of z n. When = 0, we completely understand the dynamics of z n, but when  0, the Julia set “explodes.”

2. How do you solve the equation z 2 - = 0 ??? Why the interest in these maps? 1. These are singular perturbations of z n.

Why the interest in these maps? You use Newton’s method (of course!): 1. These are singular perturbations of z n. 2. How do you solve the equation z 2 - = 0 ???

Why the interest in these maps? Iterate: 1. These are singular perturbations of z n. 2. How do you solve the equation z 2 - = 0 ??? You use Newton’s method (of course!):

Why the interest in these maps? Iterate: a singular perturbation of z/2 1. These are singular perturbations of z n. 2. How do you solve the equation z 2 - = 0 ??? You use Newton’s method (of course!):

Why the interest in these maps? Whenever the equation has a multiple root, the corresponding Newton’s method involves a singular perturbation. 1. These are singular perturbations of z n. 2. How do you solve the equation z 2 - = 0 ??? You use Newton’s method (of course!):

Why the interest in these maps? 3. We are looking at maps on the boundary of the set of rational maps of degree 2n --- a very interesting topic of contemporary research. 1. These are singular perturbations of z n. 2. How do you solve the equation z 2 - = 0 ???

Why the interest in these maps? 3. We are looking at maps on the boundary of the set of rational maps of degree 2n --- a very interesting topic of contemporary research. 1. These are singular perturbations of z n. 2. How do you solve the equation z 2 - = 0 ??? 4.And like the quadratic and exponential families, and despite the high degree of these maps, there is only one free critical orbit for these maps.

Dynamics of complex and The Julia set is: The closure of the set of repelling periodic points; The boundary of the escaping orbits; The chaotic set. The Fatou set is the complement of. A rational map of degree 2n.

When, the Julia set is the unit circle

When, the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When, the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When, the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When, the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When, the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When, the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When, the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When, the Julia set is the unit circle Colored points have orbits that escape to infinity: Escape time: red (fastest) orange yellow green blue violet (slowest)

When, the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When, the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When, the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When, the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When, the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When, the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When, the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When, the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When, the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When, the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When, the Julia set is the unit circle Black points have orbits that do not escape to infinity:

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

When, the Julia set is the unit circle The Julia set is the boundary of the black & colored regions.

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

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

Fourth reason this family is important:

2n free critical points Easy computations:

2n free critical points

Easy computations: 2n free critical points Only 2 critical values

Easy computations: 2n free critical points Only 2 critical values

Easy computations: 2n free critical points Only 2 critical values

Easy computations: 2n free critical points Only 2 critical values But really only 1 free critical orbit since the map has z  -z symmetry

Easy computations: is superattracting, so have immediate basin B mapped n-to-1 to itself. B

Easy computations: is superattracting, so have immediate basin B mapped n-to-1 to itself. B T 0 is a pole, so have trap door T mapped n-to-1 to B.

Easy computations: is superattracting, so have immediate basin B mapped n-to-1 to itself. B T So any orbit that eventually enters B must do so by passing through T. 0 is a pole, so have trap door T mapped n-to-1 to B.

The Escape Trichotomy There are three distinct ways the critical orbit can enter B: (with D. Look and D. Uminsky)

The Escape Trichotomy B is a Cantor set There are three distinct ways the critical orbit can enter B:

The Escape Trichotomy B is a Cantor set T is a Cantor set of simple closed curves There are three distinct ways the critical orbit can enter B: (this case does not occur if n = 2)

The Escape Trichotomy B is a Cantor set T is a Cantor set of simple closed curves T is a Sierpinski curve There are three distinct ways the critical orbit can enter B: (this case does not occur if n = 2)

B is a Cantor set parameter plane when n = 3 Case 1:

B is a Cantor set parameter plane when n = 3 J is a Cantor set

B is a Cantor set parameter plane when n = 3 J is a Cantor set

B is a Cantor set parameter plane when n = 3 J is a Cantor set

B is a Cantor set parameter plane when n = 3 J is a Cantor set

B is a Cantor set parameter plane when n = 3 J is a Cantor set

B is a Cantor set parameter plane when n = 3 J is a Cantor set

B is a Cantor set parameter plane when n = 3 J is a Cantor set

B is a Cantor set parameter plane when n = 3 J is a Cantor set

B is a Cantor set parameter plane when n = 3 J is a Cantor set

B is a Cantor set parameter plane when n = 3 J is a Cantor set

B is a Cantor set parameter plane when n = 3 J is a Cantor set

B is a Cantor set parameter plane when n = 3 J is a Cantor set

parameter plane when n = 3 Case 2: the critical values lie in T, not B

T parameter plane when n = 3 lies in the McMullen domain

T parameter plane when n = 3 J is a Cantor set of simple closed curves lies in the McMullen domain Remark: There is no McMullen domain in the case n = 2.

T parameter plane when n = 3 J is a Cantor set of simple closed curves lies in the McMullen domain

T parameter plane when n = 3 J is a Cantor set of simple closed curves lies in the McMullen domain

T parameter plane when n = 3 J is a Cantor set of simple closed curves lies in the McMullen domain

T parameter plane when n = 3 J is a Cantor set of simple closed curves lies in the McMullen domain

T parameter plane when n = 3 J is a Cantor set of simple closed curves lies in the McMullen domain

T parameter plane when n = 3 J is a Cantor set of simple closed curves lies in the McMullen domain

parameter plane when n = 3 Case 3: the critical orbits eventually reach T

T parameter plane when n = 3 lies in a Sierpinski hole

T parameter plane when n = 3 J is a Sierpinski curve lies in a Sierpinski hole

T parameter plane when n = 3 J is a Sierpinski curve lies in a Sierpinski hole

T parameter plane when n = 3 J is a Sierpinski curve lies in a Sierpinski hole

T parameter plane when n = 3 J is a Sierpinski curve lies in a Sierpinski hole

T parameter plane when n = 3 J is a Sierpinski curve lies in a Sierpinski hole

T parameter plane when n = 3 J is a Sierpinski curve lies in a Sierpinski hole

T parameter plane when n = 3 J is a Sierpinski curve lies in a Sierpinski hole

T parameter plane when n = 3 J is a Sierpinski curve lies in a Sierpinski hole

T parameter plane when n = 3 J is a Sierpinski curve lies in a Sierpinski hole

T parameter plane when n = 3 J is a Sierpinski curve lies in a Sierpinski hole

T parameter plane when n = 3 J is a Sierpinski curve lies in a Sierpinski hole

A Sierpinski curve is any planar set that is homeomorphic to the Sierpinski carpet fractal. The Sierpinski Carpet Three Reasons Why the Sierpinski Carpet Is the Most Important Planar Fractal

These sets occur over and over as Julia sets for these rational maps:

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”

To show that is homeomorphic to

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s Fatou set is the union of the preimages of B; all disjoint, open disks.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s Fatou set is the union of the preimages of B; all disjoint, open disks.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s If J contains an open set, then J = C.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s If J contains an open set, then J = C.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s All critical orbits escape, so the map is hyperbolic on J.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s All critical orbits escape, so the map is hyperbolic on J.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s J locally connected, so the boundaries are locally connected. Need to show they are s.c.c.’s. Can only meet at (preimages of) critical points, hence disjoint.

Need to show: compact connected nowhere dense locally connected bounded by disjoint s.c.c.’s So J is a Sierpinski curve.

Have an exact count of the number of Sierpinski holes: Theorem: Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3)

Have an exact count of the number of Sierpinski holes: Reason: The equation reduces to a polynomial of degree (n-1)(2n) (k-3) ; and it can be shown that all the roots of this polynomial are distinct. So we have exactly that many “centers” of Sierpinski holes, i.e., parameters for which the critical points all land on 0 and then on ∞. Theorem: Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3)

Have an exact count of the number of Sierpinski holes: n = 3 escape time 3 2 Sierpinski holes parameter plane n = 3 Theorem: Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3)

Have an exact count of the number of Sierpinski holes: n = 3 escape time 3 2 Sierpinski holes parameter plane n = 3 Theorem: Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3)

Have an exact count of the number of Sierpinski holes: n = 3 escape time 4 12 Sierpinski holes parameter plane n = 3 Theorem: Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3)

Have an exact count of the number of Sierpinski holes: n = 3 escape time 4 12 Sierpinski holes parameter plane n = 3 Theorem: Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3)

Have an exact count of the number of Sierpinski holes: n = 4 escape time 3 3 Sierpinski holes parameter plane n = 4 Theorem: Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3)

Have an exact count of the number of Sierpinski holes: n = 4 escape time 4 24 Sierpinski holes parameter plane n = 4 Theorem: Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3)

Have an exact count of the number of Sierpinski holes: n = 4 escape time ,653,184 Sierpinski holes Sorry. I forgot to indicate their locations. parameter plane n = 4 Theorem: Given n, there are exactly (n-1)(2n) Sierpinski holes with escape time k. (k-3)

Given two Sierpinski curve Julia sets, when do we know that the dynamics on them are the same, i.e., the maps are conjugate on the Julia sets? Main Question: These sets are homeomorphic, but are the dynamics on them the same?

Only Julia sets drawn from Sierpinski holes that are symmetrically located with respect to z   2 z, where  is an (n -1) st root of unity or complex conjugation have the same dynamics (i.e., are topologically conjugate on the J-sets) Answer:

Theorem: For any n there are exactly (n-1) (2n) Sierpinski holes with escape time k. The number of distinct conjugacy classes is given by: k-3 a. (2n) when n is odd; k-3 b. (2n) /2 + 2 when n is even. k-3k-4 Only Julia sets drawn from Sierpinski holes that are symmetrically located with respect to z   2 z, where  is an (n -1) st root of unity or complex conjugation have the same dynamics (i.e., are topologically conjugate on the J-sets) Answer:

n = 3, escape time 4, 12 Sierpinski holes, but only six conjugacy classes conjugate centers:,

n = 3, escape time 4, 12 Sierpinski holes, but only six conjugacy classes conjugate centers:,

n = 3, escape time 4, 12 Sierpinski holes, but only six conjugacy classes conjugate centers:,

n = 3, escape time 4, 12 Sierpinski holes, but only six conjugacy classes conjugate centers:,

,,,,,where n = 4, escape time 4, 24 Sierpinski holes, but only five conjugacy classes conjugate centers:

,,,,,where n = 4, escape time 4, 24 Sierpinski holes, but only five conjugacy classes conjugate centers:

,,,,,where n = 4, escape time 4, 24 Sierpinski holes, but only five conjugacy classes conjugate centers:

,,,,,where n = 4, escape time 4, 24 Sierpinski holes, but only five conjugacy classes conjugate centers:

,,,,,where n = 4, escape time 4, 24 Sierpinski holes, but only five conjugacy classes conjugate centers:

n = 4: 402,653,184 Sierpinski holes with escape time 12; 67,108,832 distinct conjugacy classes. Sorry. I again forgot to indicate their locations.

The case n = 2 vs. n > 2 Singular perturbations of z 2 + c Singular perturbations of z 3 + c A major application Websites Baby Mandelbrot sets

There are infinitely many copies of the Mandelbrot set in the parameter planes for these maps

So we see quadratic-like Julia sets in the dynamical plane

So we see quadratic-like Julia sets in the dynamical plane

So we see quadratic-like Julia sets in the dynamical plane

So we see quadratic-like Julia sets in the dynamical plane

So we see quadratic-like Julia sets in the dynamical plane

So we see quadratic-like Julia sets in the dynamical plane

But if the Mandelbrot sets are “buried”...

then the Julia sets from the main cardioid are again Sierpinski curves An attracting 3-cycle in the black regions

The case n = 2 is very different from (and much more difficult than) the case n > 2. n = 3 n = 2

One difference: there is a McMullen domain when n > 2, but no McMullen domain when n = 2 n = 3 n = 2

One difference: there is a McMullen domain when n > 2, but no McMullen domain when n = 2 n = 3 n = 2

There is lots of structure when n > 2, but what is going on when n = 2? n = 3 n = 2

There is lots of structure when n > 2, but what is going on when n = 2? n = 3 n = 2

There is lots of structure when n > 2, but what is going on when n = 2? n = 3 n = 2

Also, not much is happening for the Julia sets near 0 when n > 2 n = 3

The Julia set is always a Cantor set of circles. n = 3

The Julia set is always a Cantor set of circles.

The Julia set is always a Cantor set of circles. There is always a round annulus of some fixed width in the Fatou set, so the Julia set does not converge to the unit disk.

n = 2 But when n = 2, lots of things happen near the origin; in fact, the Julia sets converge to the unit disk as disk-converge

For the family the Julia sets again converge to the unit disk, but only if  0 along n - 1 special rays. (with M. Morabito) n = 6 n = 4

Consider the family of maps where c is the center of a hyperbolic component of the Mandelbrot set. c = 0

Consider the family of maps where c is the center of a hyperbolic component of the Mandelbrot set. c = -1

Consider the family of maps where c is the center of a hyperbolic component of the Mandelbrot set. c = i

When, the Julia set again expodes and converges to the filled Julia set for z 2 + c. (with R. Kozma)

When, the Julia set again expodes and converges to the filled Julia set for z 2 + c. (with R. Kozma)

When, the Julia set again expodes and converges to the filled Julia set for z 2 + c. (with R. Kozma)

When, the Julia set again expodes and converges to the filled Julia set for z 2 + c. (with R. Kozma)

When, the Julia set again expodes and converges to the filled Julia set for z 2 + c. (with R. Kozma) An inverted Douady rabbit

If you chop off the “ears” of each internal rabbit in each component of the original Fatou set, then what’s left is another Sierpinski curve (provided that both of the critical orbits eventually escape).

The case n > 2 is also very different: (E. Russell)

When is small, the Julia set contains a Cantor set of “circles” surrounding the origin.....

infinitely many of which are “decorated” and there are also Cantor sets of buried points

Here’s the parameter plane when n = 2:

Rotate it by 90 degrees: You will win a Fiends Medal if you.....

... help me figure out who this person is!

Paul Blanchard Toni Garijo Matt Holzer Robert Kozma Dan Look Sebastian Marotta Mark Morabito with: Monica Moreno Rocha Kevin Pilgrim Elizabeth Russell Yakov Shapiro David Uminsky Sum Wun Ellce

Paul Blanchard Toni Garijo Matt Holzer Robert Kozma Dan Look Sebastian Marotta Mark Morabito with: Monica Moreno Rocha Kevin Pilgrim Elizabeth Russell Yakov Shapiro David Uminsky Sum Wun Ellce

Paul Blanchard Toni Garijo Matt Holzer Robert Kozma Dan Look Sebastian Marotta Mark Morabito suspects: Monica Moreno Rocha Kevin Pilgrim Elizabeth Russell Yakov Shapiro David Uminsky Sum Wun Ellce

Websites: math.bu.edu/DYSYS Mandelbrot set Explorer java applets for quadratic polynomials exponentials, sines, etc. rational maps math.bu.edu/people/bob recent papers