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Small Perturbations of z n Daniel Cuzzocreo Boston University Summer Conference on Topology and its Applications Nipissing University July 24, 2013 Small Perturbations of z n Daniel Cuzzocreo ( Boston University) July 24, 2013 1 Joint work

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Small Perturbations of zn

Daniel Cuzzocreo

Boston University

Summer Conference on Topology and its Applications Nipissing University July 24, 2013

Daniel Cuzzocreo ( Boston University) Small Perturbations of zn July 24, 2013 1

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Joint work with Bob Devaney

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Basic Notions: Complex Dynamics

In Complex Dynamics, we consider the behavior of points under iteration of a holomorphic function. In this setting, f : ˆ C → ˆ C will be a rational map.

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Basic Notions: Complex Dynamics

In Complex Dynamics, we consider the behavior of points under iteration of a holomorphic function. In this setting, f : ˆ C → ˆ C will be a rational map.

Definitions

For a point z ∈ ˆ C, the sequence (z, f(z), f 2(z), . . . ) is called the

  • rbit of z under f.

If z = f n(z) for some n, with n minimal, then we say z is periodic, with period n. In this case, the complex number λ = (f n)′(z) is called the multiplier of z.

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Julia and Fatou Sets

For a periodic point z, we say z is: attracting if |λ| < 1 superattracting if λ = 0 repelling if |λ| > 1 indifferent if |λ| = 1

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Julia and Fatou Sets

For a periodic point z, we say z is: attracting if |λ| < 1 superattracting if λ = 0 repelling if |λ| > 1 indifferent if |λ| = 1 This gives a natural partition of the Riemann sphere: The Julia set, J(f), is the closure of the set of repelling periodic points.

Dynamics of f on the Julia set are "chaotic."

The Fatou set is the complement of the Julia set.

Dynamics of f on the Fatou set are "stable."

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Examples: Quadratic Polynomials

f(z) = z2 "The Unit Circle" f(z) = z2 − 1 "The Basilica" The Julia set is the boundary between the black and orange regions.

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Introduction: Perturbed Polynomials

We consider the singularly perturbed polynomial map Fλ: Fλ(z) = zn + λ zd

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Introduction: Perturbed Polynomials

We consider the singularly perturbed polynomial map Fλ: Fλ(z) = zn + λ zd Usually n, d ≥ 2. Often (but not always) we take n = d for added symmetry.

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Introduction: Perturbed Polynomials

We consider the singularly perturbed polynomial map Fλ: Fλ(z) = zn + λ zd Usually n, d ≥ 2. Often (but not always) we take n = d for added symmetry. For λ = 0 this map is the complex polynomial z → zn. When λ = 0 we have replaced the superattracting fixed point at the

  • rigin with a pole.

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Motivation

Why study these maps?

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Motivation

Why study these maps? Allows us to study rational maps of arbitrarily high degree. Many important features in the case n = d = 3, e.g., persist in all higher degrees.

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Motivation

Why study these maps? Allows us to study rational maps of arbitrarily high degree. Many important features in the case n = d = 3, e.g., persist in all higher degrees. As λ → 0, we approach the boundary of Ratn+d, the space of rational maps of degree n + d. The structure of these spaces is a very active area of research.

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Motivation

Why study these maps? Allows us to study rational maps of arbitrarily high degree. Many important features in the case n = d = 3, e.g., persist in all higher degrees. As λ → 0, we approach the boundary of Ratn+d, the space of rational maps of degree n + d. The structure of these spaces is a very active area of research. Symmetries always allow us to study a natural one parameter family in any degree. There is always a single "free" critical orbit.

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Motivation

Why study these maps? Allows us to study rational maps of arbitrarily high degree. Many important features in the case n = d = 3, e.g., persist in all higher degrees. As λ → 0, we approach the boundary of Ratn+d, the space of rational maps of degree n + d. The structure of these spaces is a very active area of research. Symmetries always allow us to study a natural one parameter family in any degree. There is always a single "free" critical orbit. Interesting dynamical behavior and topological features. Sierpi´ nski curve Julia sets are extremely common, for example.

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Motivation

Why study these maps? Allows us to study rational maps of arbitrarily high degree. Many important features in the case n = d = 3, e.g., persist in all higher degrees. As λ → 0, we approach the boundary of Ratn+d, the space of rational maps of degree n + d. The structure of these spaces is a very active area of research. Symmetries always allow us to study a natural one parameter family in any degree. There is always a single "free" critical orbit. Interesting dynamical behavior and topological features. Sierpi´ nski curve Julia sets are extremely common, for example.

Daniel Cuzzocreo ( Boston University) Small Perturbations of zn July 24, 2013 7

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Preliminaries

For this talk, we are interested in the case where |λ| is small. The dynamics here are well understood when n and d are not both 2, but much more complicated when n = d = 2.

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Preliminaries

For this talk, we are interested in the case where |λ| is small. The dynamics here are well understood when n and d are not both 2, but much more complicated when n = d = 2. For simplicity, we’ll assume n = d, so that our map is Fλ(z) = zn + λ zn , n ≥ 2

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Preliminaries

For this talk, we are interested in the case where |λ| is small. The dynamics here are well understood when n and d are not both 2, but much more complicated when n = d = 2. For simplicity, we’ll assume n = d, so that our map is Fλ(z) = zn + λ zn , n ≥ 2 ∞ is always a superattracting fixed point.

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Preliminaries

For this talk, we are interested in the case where |λ| is small. The dynamics here are well understood when n and d are not both 2, but much more complicated when n = d = 2. For simplicity, we’ll assume n = d, so that our map is Fλ(z) = zn + λ zn , n ≥ 2 ∞ is always a superattracting fixed point. Only pole is at 0, which is also a critical point.

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Preliminaries

For this talk, we are interested in the case where |λ| is small. The dynamics here are well understood when n and d are not both 2, but much more complicated when n = d = 2. For simplicity, we’ll assume n = d, so that our map is Fλ(z) = zn + λ zn , n ≥ 2 ∞ is always a superattracting fixed point. Only pole is at 0, which is also a critical point. 2n other critical points lie at

2n

√ λ.

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Preliminaries

For this talk, we are interested in the case where |λ| is small. The dynamics here are well understood when n and d are not both 2, but much more complicated when n = d = 2. For simplicity, we’ll assume n = d, so that our map is Fλ(z) = zn + λ zn , n ≥ 2 ∞ is always a superattracting fixed point. Only pole is at 0, which is also a critical point. 2n other critical points lie at

2n

√ λ. These map to two critical values at ±2 √ λ

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Preliminaries

We denote the immediate basin of ∞ by Bλ, and the connected component of the basin of ∞ which contains 0 by Tλ (the "trap door"). These sets may coincide.

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Preliminaries

We denote the immediate basin of ∞ by Bλ, and the connected component of the basin of ∞ which contains 0 by Tλ (the "trap door"). These sets may coincide.

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The Escape Trichotomy

The behavior of the critical points determines the topology of the Julia set of Fλ:

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The Escape Trichotomy

The behavior of the critical points determines the topology of the Julia set of Fλ:

The Escape Trichotomy (Devaney, Look, Uminsky; 2005)

Let vλ = Fλ(cλ) be a critical value, and suppose F k

λ (cλ) → ∞. Then:

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The Escape Trichotomy

The behavior of the critical points determines the topology of the Julia set of Fλ:

The Escape Trichotomy (Devaney, Look, Uminsky; 2005)

Let vλ = Fλ(cλ) be a critical value, and suppose F k

λ (cλ) → ∞. Then:

1

if vλ lies in Bλ, then J(Fλ) is a Cantor set;

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The Escape Trichotomy

The behavior of the critical points determines the topology of the Julia set of Fλ:

The Escape Trichotomy (Devaney, Look, Uminsky; 2005)

Let vλ = Fλ(cλ) be a critical value, and suppose F k

λ (cλ) → ∞. Then:

1

if vλ lies in Bλ, then J(Fλ) is a Cantor set;

2

if vλ lies in Tλ = Bλ, then J(Fλ) is a Cantor set of concentric simple closed curves surrounding the origin;

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The Escape Trichotomy

The behavior of the critical points determines the topology of the Julia set of Fλ:

The Escape Trichotomy (Devaney, Look, Uminsky; 2005)

Let vλ = Fλ(cλ) be a critical value, and suppose F k

λ (cλ) → ∞. Then:

1

if vλ lies in Bλ, then J(Fλ) is a Cantor set;

2

if vλ lies in Tλ = Bλ, then J(Fλ) is a Cantor set of concentric simple closed curves surrounding the origin;

3

in all other cases, J(Fλ) is a connected set.

In particular, if F j

λ(vλ) ∈ Tλ = Bλ for some j ≥ 1, then J(Fλ) is a

Sierpi´ nski curve (i.e., homeomorphic to the Sierpi´ nski carpet). Sets

  • f λ values where this occurs are called Sierpi´

nski holes.

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Example: Case 1, n=3

A Cantor set.

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Example: Case 2, n=3

A Cantor set of simple closed curves.

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Example: Case 3, n=3

A Sierpi´ nski curve.

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Small perturbations: n ≥ 3

What happens when the parameter is very small? The answer is very different for the cases n = 2 and n ≥ 3.

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Small perturbations: n ≥ 3

What happens when the parameter is very small? The answer is very different for the cases n = 2 and n ≥ 3. When n ≥ 3, there exists a punctured neighborhood of λ = 0 for which Case 2 occurs, (the McMullen domain). The critical values both lie in the trap door and the Julia set is a Cantor set of simple closed curves.

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Small perturbations: n ≥ 3

What happens when the parameter is very small? The answer is very different for the cases n = 2 and n ≥ 3. When n ≥ 3, there exists a punctured neighborhood of λ = 0 for which Case 2 occurs, (the McMullen domain). The critical values both lie in the trap door and the Julia set is a Cantor set of simple closed curves.

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Small perturbations: n ≥ 3

Moreover, as a function of λ, the width of the largest annulus in the Fatou set is bounded away from zero (Devaney, Garijo; 2006).

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Small perturbations: n ≥ 3

Moreover, as a function of λ, the width of the largest annulus in the Fatou set is bounded away from zero (Devaney, Garijo; 2006). n = 3, λ ≈ −0.005 n = 3, λ ≈ 10−6 There is always at least one "thick" annulus in the Fatou set as λ → 0.

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Small perturbations: n=2

When n = 2, no McMullen domain exists. Why?

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Small perturbations: n=2

When n = 2, no McMullen domain exists. Why? If the critical values are to lie in the trap door, the Riemann-Hurwitz formula requires that the preimage of the trap door must be a single annulus X mapped 2n-to-1 onto Tλ. The remaining annuli A1 and A2 are each mapped n-to-1

  • nto A = A1 ∪ X ∪ A2.

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Small perturbations: n=2

When n = 2, no McMullen domain exists. Why? An n-to-1 covering map expands the modulus of an annulus by a factor of n, so when n = 2 we have

1 2mod A = mod A1 = mod A2.

But mod A = mod A1 + mod X + mod A2, so there is no room for X.

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Small perturbations, n=2

Since we have no McMullen domain, for small λ, J(Fλ) is connected by the Escape Trichotomy.

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Small perturbations, n=2

Since we have no McMullen domain, for small λ, J(Fλ) is connected by the Escape Trichotomy. Yet there are uncountably many conjugacy classes of maps in any neighborhood of λ = 0.

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Small perturbations, n=2

Since we have no McMullen domain, for small λ, J(Fλ) is connected by the Escape Trichotomy. Yet there are uncountably many conjugacy classes of maps in any neighborhood of λ = 0. Moreover, we have the following theorem:

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Small perturbations, n=2

Since we have no McMullen domain, for small λ, J(Fλ) is connected by the Escape Trichotomy. Yet there are uncountably many conjugacy classes of maps in any neighborhood of λ = 0. Moreover, we have the following theorem:

Theorem (Devaney, Garijo; 2006)

As λ approaches zero, the Julia set for the map Fλ(z) = z2 + λ z2 converges to the closed unit disk in the Hausdorff metric.

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Small perturbations, n=2

As λ gets small, bounded components of the Fatou set shrink.

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Small perturbations, n=2

Proof Idea

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Small perturbations, n=2

Proof Idea

If the conclusion fails to hold, then for all ǫ sufficiently small, there exists a sequence λj converging to zero, and a corresponding sequence zj in the closed disk such that Bǫ(zj) lies in the Fatou set of Fλj for all j.

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Small perturbations, n=2

Proof Idea

If the conclusion fails to hold, then for all ǫ sufficiently small, there exists a sequence λj converging to zero, and a corresponding sequence zj in the closed disk such that Bǫ(zj) lies in the Fatou set of Fλj for all j. By compactness, there is a subsequence of the zj converging to a point z∗ in D, so we may assume wlog that Bǫ(z∗) is in the Fatou set of Fλj for all j.

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Small perturbations, n=2

Proof Idea

If the conclusion fails to hold, then for all ǫ sufficiently small, there exists a sequence λj converging to zero, and a corresponding sequence zj in the closed disk such that Bǫ(zj) lies in the Fatou set of Fλj for all j. By compactness, there is a subsequence of the zj converging to a point z∗ in D, so we may assume wlog that Bǫ(z∗) is in the Fatou set of Fλj for all j. Since λj → 0, Fλj ≈ z2 for large j, so for large k, F k

λj wraps Bǫ(z∗)

around the origin, disconnecting the Julia set by forward invariance.

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Parameter space, n=2

Hence the structure of the parameter plane near λ = 0 for n = 2 is quite complicated.

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Parameter space, n=2

Hence the structure of the parameter plane near λ = 0 for n = 2 is quite complicated.

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Parameter space, n=2

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Parameter space, n=2

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Parameter space, n=2

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Parameter space, n=2

Certain structures are clearly visible however:

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Parameter space, n=2

Certain structures are clearly visible however: We can see many "rings" of Sierpi´ nski holes alternating with what appear to be baby Mandelbrot sets.

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Parameter space, n=2

Certain structures are clearly visible however: We can see many "rings" of Sierpi´ nski holes alternating with what appear to be baby Mandelbrot sets. Looking more closely at any such hole reveals many more such rings surrounding it.

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Parameter space, n=2

Certain structures are clearly visible however: We can see many "rings" of Sierpi´ nski holes alternating with what appear to be baby Mandelbrot sets. Looking more closely at any such hole reveals many more such rings surrounding it. Our goal is ultimately to describe this structure completely.

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Mandelpi´ nski Necklaces

In the dynamical plane, all parameters within the "dividing circle" of radius 1/16 have the property that |vλ| < |cλ|. There exist concentric curves Ck for all integers k such that:

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Mandelpi´ nski Necklaces

In the dynamical plane, all parameters within the "dividing circle" of radius 1/16 have the property that |vλ| < |cλ|. There exist concentric curves Ck for all integers k such that: C0 is defined to be the critical circle of radius

4

  • |λ|.

Ck+1 surrounds and maps 2-1 onto Ck for k > 0, and C−k lies inside the critical circle and maps 2-1 onto Ck−1 for k > 0.

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Mandelpi´ nski Necklaces

In the dynamical plane, all parameters within the "dividing circle" of radius 1/16 have the property that |vλ| < |cλ|. There exist concentric curves Ck for all integers k such that: C0 is defined to be the critical circle of radius

4

  • |λ|.

Ck+1 surrounds and maps 2-1 onto Ck for k > 0, and C−k lies inside the critical circle and maps 2-1 onto Ck−1 for k > 0. The kth Mandelpi´ nski necklace in the dynamical plane is a simple closed curve of parameters for which the critical values lie on C−k.

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Example: A Mandelpi´ nski Necklace for n=2

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Sub-necklaces

In the dynamical plane, the sector of points whose arguments lie between two adjacent critical points is mapped 1-1 onto the complement of the rays extending from the critical values to infinity. This sector therefore contains a preimage of the trap door, as well as

  • f all C−k such that vλ lies outside C−k.

Thus the preimage of the trap door is surrounded by infinitely many simple closed curves that map onto these C−k’s. This recursively yields a structure of sub-rings and sub-sub-rings in the dynamical plane that appears to be replicated in parameter space. Can we explicitly show that this structure persists in the parameter plane at each level? (Work in progress).

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Further Questions

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Further Questions

Can we explicitly describe the arrangement of all Sierpi´ nski holes in some neighborhood of λ = 0?

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Further Questions

Can we explicitly describe the arrangement of all Sierpi´ nski holes in some neighborhood of λ = 0? What formula gives the number of Sierpi´ nski holes in each sub-necklace (and sub-sub-necklace, etc.)?

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Further Questions

Can we explicitly describe the arrangement of all Sierpi´ nski holes in some neighborhood of λ = 0? What formula gives the number of Sierpi´ nski holes in each sub-necklace (and sub-sub-necklace, etc.)? Can we prove the existence of all baby Mandelbrot sets in each necklace?

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Thanks

Thank you for your attention.

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