Critical Portraits of Complex Polynomials John C. Mayer Department - - PowerPoint PPT Presentation

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Critical Portraits of Complex Polynomials John C. Mayer Department - - PowerPoint PPT Presentation

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations Critical Portraits of Complex Polynomials John C. Mayer Department of Mathematics University of Alabama at Birmingham

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Critical Portraits of Complex Polynomials

John C. Mayer

Department of Mathematics University of Alabama at Birmingham

May 24, 2018 2018 Topology Workshop Nipissing University

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Coworkers

PhD Dissertation (2015): On the Simplest Lamination of a Given Identity Return Triangle Brandon L. Barry UG Posters: (2017) Critical Portraits and Weakly BiColored Trees (2018) Ambiguous or Non-Generic Critical Portraits of Complex Polynomials David J. George and Simon D. Harris MS Thesis (2017): Exponential Laminations Patrick B. Hartley

Work in Progress

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Outline

1

Polynomial Julia Sets and Laminations

2

Critical Chords and Pullbacks

3

Critical Portraits, Dual Graphs, and Simplest Laminations

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Outline

1

Polynomial Julia Sets and Laminations

2

Critical Chords and Pullbacks

3

Critical Portraits, Dual Graphs, and Simplest Laminations

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Outline

1

Polynomial Julia Sets and Laminations

2

Critical Chords and Pullbacks

3

Critical Portraits, Dual Graphs, and Simplest Laminations

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Complex Polynomials

Polynomial P : C → C of degree d ≥ 2: P(z) = adzd + ad−1zd−1 + · · · + a1z + a0 Compactify C to C∞. For P, ∞ is attracting fixed point: for z ∈ C with |z| sufficiently large, limn→∞ Pn(z) = ∞. Basin of attraction of ∞: B∞ := {z ∈ C | lim

n→∞ Pn(z) = ∞}

B∞ is an open set.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Julia and Fatou Sets

Definitions: Julia set J(P) := boundary of B∞. Fatou set F(P) := C∞ \ J(P). Filled Julia set K(P) := C∞ \ B∞. Fun Facts: J(P) is nonempty, compact, and perfect. K(P) does not separate C. Attracting orbits are in Fatou set. Repelling orbits are in Julia set. We will assume J(P) is connected (a continuum: compact, connected metric space).

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Basillica z → z2 − 1

Julia set pictures by Fractalstream

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Douady Rabbit z → z2 + (−0.12 + 0.78i)

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Twisted Rabbit z → z2 + (0.057 + 0.713i)

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Airplane z → z2 − 1.75

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Minnie Mouse z → z3 + (0.545 + 0.539i)

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Helicopter z → z3 + (−0.2634 − 1.2594i)

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Scorpion/Scepter z → z3 + 3(0.785415i)z2

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Butterfly z → z3 + 3(−0.5)z2 + (0.75 + 0.661438i)

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Ninja Throwing Star z → z3 + (0.20257 + 1.095i)

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

The Simplest Julia Set – the Unit Circle ∂D

P(z) = z2 re2πi t → r 2e2πi 2t

The complement C∞ \ D of the closed unit disk is the basin of attraction, B∞, of infinity.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Dynamics on the Unit Circle

Consider P(z) = zd on the unit circle ∂D. z = re2πt → r de2πi(dt) − → Angle 2πt → 2π(dt). Measure angles in revolutions: Points on ∂D are coordinatized by [0, 1). σd : t → dt (mod 1) on ∂D . Example d = 2:

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Bottcher’s Theorem

D∞ D∞ B∞ B∞

✲

z→zd

❄

φ

❄

φ

✲

P

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

External Rays P(z) = z2 + (−0.12 + 0.78i)

1/7 2/7 4/7

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

External Rays − → Laminations

Laminations were introduced by William Thurston as a way

  • f encoding connected polynomial Julia sets.

Coincident external rays Rabbit triangle

1/7 2/7 4/7 21 / 87

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

The Rabbit Lamination

The rabbit Julia set The rabbit lamination

1/7 2/7 4/7

Hyperbolic lamination pictures courtesy of Logan Hoehn

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Laminations of the Unit Disk D

Definition A lamination L is a collections of chords of D, which we call leaves, with the property that any two leaves meet, if at all, in a point of ∂D, and such that L has the property that L∗ := ∂D ∪ {∪L} is a closed subset of D. We allow degenerate leaves – points of ∂D. Note that L∗ is a continuum: compact, connected metric space.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Extending σd to Leaves

If ℓ ∈ L is a leaf, we write ℓ = ab, where a and b are the endpoints of ℓ in ∂D. We define σd(ℓ) to be the chord σd(a)σd(b). The length of a chord is the length of the shorter arc of the circle subtended. If it happens that σd(a) = σd(b), then σd(ℓ) is a point, called a critical value of L, and we say ℓ is a critical leaf.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Making the Lamination dynamic!

Definition (Sibling Invariant Lamination) A lamination L is said to be sibling d-invariant provided that:

1

(Forward Invariant) For every ℓ ∈ L, σd(ℓ) ∈ L.

2

(Backward Invariant) For every non-degenerate ℓ′ ∈ L, there is a leaf ℓ ∈ L such that σd(ℓ) = ℓ′.

3

(Sibling Invariant) For every ℓ1 ∈ L with σd(ℓ1) = ℓ′, a non-degenerate leaf, there is a full sibling collection of pairwise disjoint leaves {ℓ1, ℓ2, . . . , ℓd} ⊂ L such that σd(ℓi) = ℓ′. Conditions (1), (2) and (3) allow generating a sibling invariant lamination from a finite amount of initial data.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

σ2 Binary Coordinates Location dynamically defined. 1

00 01 10 11

000 001 010 111 110 011 100 101

0000 0001

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

σ2 Binary Coordinates and Rabbit

In binary coordinates, σ2 is the “forgetful” shift. The overline means the coordinates repeat.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Generating a Lamination from Finite Initial Data

Definition (Pullback Scheme) A pullback scheme for σd is a collection of d branches τ1, τ2, . . . , τd of the inverse of σd whose ranges partition ∂D.

001 010 100

Data: Forward invariant lamination.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Scheme

Definition (Guiding Critical Chords) The generating data of a pullback scheme are a forward invariant periodic collection of leaves and a collection of d interior disjoint guiding critical chords.

001 010 100 001 010 100 1010

Data: Forward invariant lamination. Guiding critical chord(s).

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

001 010 100

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

001 010 100 001 010 100 1010 1100 0001

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Importance of Guiding Crital Chord

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Ambiguity

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Quadratic Lamination and Julia Set

Rabbit Lamination Rabbit Julia Set Quotient space in plane = ⇒ homeomorphic to rabbit Julia set. Semiconjugate dynamics

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Lamination Data for Rabbit Lamination

The critical chord and one endpoint determine the lamination.

001 010 100 1010

The rabbit triangle’s vertices are the only periodic orbit that stays in the left half. The fixed point 0 is the only periodic orbit that stays in the right half.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Critical Portrait − → Dual Graph

Abstract from the Lamination Data just the critical chord. Bicolored Critical Portrait Bicolored Dual Graph

P F

P F

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Cast of Characters

Julia Set Lamination Lamination Data → →

001 010 100 1010

→ →

P F

→

P F

Critical Portrait Bicolored Tree

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Cast of Characters

Julia Set Lamination Lamination Data ↔ ↔

001 010 100 1010

↔ →

P F

↔

P F

Critical Portrait Bicolored Tree

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

The Simplest Lamination

Definition For given lamination data for σd consisting of a collection of periodic polygons and guiding critical chords, we call a pull-back lamination whose Fatou domains (1) are bordered by sides of the given polygons, and (2) contain the guiding critical chords, a simplest lamination for the given data. There is no claim that a simplest lamination is unique, though that would be a desireable consequence of a good definition. See Brandon Barry’s dissertation:

  • Theorem. For σ3, there is always a simplest lamination.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

σ3 and Ternary Coordinates

Ternary coordinates correspond to shift σ3.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Critical Chords, Critical Sectors, and Fixed Points

Example for σ3 (angle-tripling):

P F 1 P

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Weakly Bicolored Trees

Theorem: Critical portraits correspond dually to weakly bicolored trees. [George, Harris] Definition A tree is said to be weakly bicolored provided it satisfies the following conditions:

1

Each of two vertex colors (say, red and blue) is used at least once.

2

One vertex color, say blue, can be adjacent to itself.

3

One vertex color, say red, cannot be adjacent to itself. Problem: How many different weakly bicolored trees are there, up to orientation-preserving planar isomorphism, with n vertices?

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Weakly Bicolored Trees

Below are the three possible weakly bicolored trees on three vertices up to orientation-preserving planar isomorphism:

F P F F P F P F P

Graphs corresponding dually to critical portraits are always trees. Critical portraits that produce equivalent laminations are rotations of each other.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: P-F-F

Bicolored Tree Critical Portrait Lamination Data

F P F →

1 P F F

→ → → → Lamination Julia Set

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: P-F-F

Bicolored Tree Critical Portrait Lamination Data

F P F →

1 P F F

→

001 010 100 1010 2010 1

→ → → Lamination Julia Set

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

First two Pullbacks of Triangle

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

First two Pullbacks of Triangle

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: P-F-F

Bicolored Tree Critical Portrait Lamination Data

F P F →

1 P F F

→

001 010 100 1010 2010 1

→ → → Lamination Julia Set

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Ternary Checkerboard

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Verifying Angles

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Verifying Angles

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: P-F-P

Bicolored Tree Critical Portrait Lamination Data

P F P →

P P F

→

P P F 1 12 21

→ → → Lamination Julia Set

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pullback Sequence

Scorpion Lamination Scorpion Julia set

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: P-F-P

Bicolored Tree Critical Portrait Lamination Data

P F P →

P P F

→

P P F 1 12 21

→ → → Lamination Julia Set

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Finding the Diamond

Can we find a cubic polynomial and a resulting Julia set incorporating the diamond?

1 01 10 02 20

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Finding the Diamond

The lamination data is enough to find the diamond Julia set among a parameterized family of cubic polynomials: z → z3 + 3az2 + b, for (a, b) ∈ C2. Two period 2 Fatou domains − → Two period 2 critical points − → 0 and −2a − → Two simultaneous equations in parameters a and b − → Multiple specific parameters (a, b): set a to -0.5. set b to 0.75 + 0.661438i. set a to -0.5. set b to 0.75 - 0.661438i. set a to 0.5. set b to -0.75 + 0.661438i. set a to 0.5. set b to -0.75 - 0.661438i.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Finding the Diamond

The lamination data is enough to find the diamond Julia set among a parameterized family of cubic polynomials: z → z3 + 3az2 + b, for (a, b) ∈ C2. Two period 2 Fatou domains − → Two period 2 critical points − → 0 and −2a − → Two simultaneous equations in parameters a and b − → Multiple specific parameters (a, b): set a to -0.5. set b to 0.75 + 0.661438i. set a to -0.5. set b to 0.75 - 0.661438i. set a to 0.5. set b to -0.75 + 0.661438i. set a to 0.5. set b to -0.75 - 0.661438i.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: F-P-F (Diamond)

Bicolored Tree Critical Portrait Lamination Data

F P F→

P F 1 F

→

1 01 10 02 20

→ → → Lamination Julia Set

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: F-P-F (Diamond)

Bicolored Tree Critical Portrait Lamination Data

F P F→

P F 1 F

→

1 01 10 02 20

→ → → Lamination Julia Set (Butterfly)

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: F-P-F (Diamond)

Bicolored Tree Critical Portrait Lamination Data

F P F→

P F 1 F

→

1 01 10 02 20

→ → → Lamination Julia Set (Butterfly)

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Periodic Lamination Data

120 212 122 201 221 012

What about guiding critical chords? cf: Brandon Barry

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Periodic Lamination Data

201 120 212 122 221 012 1

Consider fixed points and chord closest to critical length.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: ?-?-?

Bicolored Tree Critical Portrait Periodic Lamination Data →

? F 1 ?

→

201 120 212 122 221 012 1

→ → → Lamination Julia Set

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: F-F-? or F-?-F

Bicolored Tree Critical Portrait Periodic Lamination Data

F P F

→

P F F 1

→

201 120 212 122 221 012 1

→ → → Lamination Julia Set

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: F-F-P

Bicolored Tree Critical Portrait Lamination Data

F P F

→

P F F 1

→

120 212 122 201 221 012 0201 2201

→ → → Lamination Julia Set

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Pulling back longest leaf

120 212 0201 2201 122 201 M m M' M''

G

0122 1122 020

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: F-F-P

Bicolored Tree Critical Portrait Lamination Data

F P F

→

P F F 1

→

120 212 122 201 221 012 0201 2201

→ → → Lamination Helicopter Julia Set

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Periodic Forcing

120 212 122 201 221 012 0201 2201 1 12 21

A major goal is to understand periodic forcing for degree d ≥ 3.

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Some Problems

1

What role is played by periodic forcing in determining the simplest lamination from given periodic data.

2

Does each initial lamination data set (periodic polygons and critical chords) correspond to some complex polynomial?

3

How many weakly bicolored trees are there for a given degree (number of vertices)?

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Meta-Problems

1

Are laminations useful in understanding polynomial dynamics?

Wandering branch points exist for polynomial Julia sets of degree 3. [Blokh and Oversteegen] There are two distinct kinds of branch points that first return without rotation for polynomial Julia sets of degree 3. [Barry and M.]

2

Are laminations applicable outside polynomial dynamics?

Julia sets of the exponential family Eλ(z) = λez can be described by laminations of the half-plane. [Hartley and M.]

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Meta-Problems

1

Are laminations useful in understanding polynomial dynamics?

Wandering branch points exist for polynomial Julia sets of degree 3. [Blokh and Oversteegen] There are two distinct kinds of branch points that first return without rotation for polynomial Julia sets of degree 3. [Barry and M.]

2

Are laminations applicable outside polynomial dynamics?

Julia sets of the exponential family Eλ(z) = λez can be described by laminations of the half-plane. [Hartley and M.]

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Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Meta-Problems

1

Are laminations useful in understanding polynomial dynamics?

Wandering branch points exist for polynomial Julia sets of degree 3. [Blokh and Oversteegen] There are two distinct kinds of branch points that first return without rotation for polynomial Julia sets of degree 3. [Barry and M.]

2

Are laminations applicable outside polynomial dynamics?

Julia sets of the exponential family Eλ(z) = λez can be described by laminations of the half-plane. [Hartley and M.]

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SLIDE 83

Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Meta-Problems

1

Are laminations useful in understanding polynomial dynamics?

Wandering branch points exist for polynomial Julia sets of degree 3. [Blokh and Oversteegen] There are two distinct kinds of branch points that first return without rotation for polynomial Julia sets of degree 3. [Barry and M.]

2

Are laminations applicable outside polynomial dynamics?

Julia sets of the exponential family Eλ(z) = λez can be described by laminations of the half-plane. [Hartley and M.]

83 / 87

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SLIDE 84

Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Meta-Problems

1

Are laminations useful in understanding polynomial dynamics?

Wandering branch points exist for polynomial Julia sets of degree 3. [Blokh and Oversteegen] There are two distinct kinds of branch points that first return without rotation for polynomial Julia sets of degree 3. [Barry and M.]

2

Are laminations applicable outside polynomial dynamics?

Julia sets of the exponential family Eλ(z) = λez can be described by laminations of the half-plane. [Hartley and M.]

84 / 87

slide-85
SLIDE 85

Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Preview of David George’s talk

Correspondence between generic critical portraits and bicolored trees. Non-generic critical portraits, all-critical polygons, and tricolored trees. Orbits under σd commute with rotation by a fixed point. The pullback step under σd commutes with rotation by a fixed point. Dynamical equivalence of pullback laminations.

THANKS!

85 / 87

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SLIDE 86

Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Rotation by a Fixed Point: F-F-P − → P-F-F

F P F

Bicolored Tree Helicopter Julia Set

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SLIDE 87

Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Rotation by a Fixed Point: F-F-P − → P-F-F

F P F

Rotated Bicolored Tree Rotated Helicopter

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SLIDE 88

Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Role Reversal: F-P-F

Bicolored Tree Critical Portrait Lamination Data

F P F→

P F 1 F

→

1 02 20

→ → → Lamination Julia Set

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SLIDE 89

Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations

Exponential Laminations

Julia set of Eλ(z) = λez, λ = 3 + π Half-plane lamination Laminations can be adapted to the Exponential family of functions using a half-plane model. Cf: Hartley

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