the lavaurs algorithm for thurston's quadratic minor

From Julia Set to LaminationQuadratic Polynomials and Quadratic Parameter Space

Preview: Cubic Polynomials

The Lavaurs Algorithm for Thurston’sQuadratic Minor Lamination

John C. Mayer

Department of MathematicsUniversity of Alabama at Birmingham

May 25, 2015

Nipissing Topology WorkshopMay 25-29, 2015

North Bay, Ontairo

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Outline

1 From Julia Set to Lamination

2 Quadratic Polynomials and Quadratic Parameter Space

3 Preview: Cubic Polynomials

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Outline




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Outline




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The Simplest Julia Set – the Unit CircleP(z) = z2 re2πi t 7→ r 2e2πi 2t

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

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Dynamics on the Circle

Consider special case P(z) = zd on the unit circle ∂D.z = re2πt 7→ rde2πi(dt).Angle 2πt 7→ 2π(dt).“Forget” 2π: then t 7→ dt (mod 1) on ∂D.We measure angles in revolutions.Points on ∂D are coordinatized by [0,1).For a real number s,

s (mod 1)

is the fractional part of s.

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s (mod 1)


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s (mod 1)


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σd Dynamics on the Circle

How points move under angle-doublingBy σd we denote the map

σd(t) = dt (mod 1)

defined on the circle, ∂D, parameterized by [0,1).

σ2 : t 7→ 2t (mod 1), angle-doubling, corresponds toz 7→ z2.

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σd(t) = dt (mod 1)

defined on the circle, ∂D, parameterized by [0,1).

σ2 : t 7→ 2t (mod 1), angle-doubling, corresponds toz 7→ z2.

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σd(t) = dt (mod 1)

defined on the circle, S, parameterized by [0,1).

σ2 : t 7→ 2t (mod 1), angle-doubling.

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The Rabbit Julia SetP(z) = z2 + (−0.12 + 0.78i)

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Bottkher’s Theorem

By D∞, “the disk at infinity,” we mean C∞ \ D, the complementof the closed unit disk.

Theorem (Bottkher)Let P be a polynomial of degree d. If the filled Julia set K isconnected, then there is a conformal isomorphism

φ : D∞ → B∞,

tangent to the identity at∞, that conjugates P to z → zd .

Laminations were introduced by William Thurston as a wayof encoding connected polynomial Julia sets.

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D∞ D∞

B∞ B∞

-z 7→z2

?

φ

?

φ

-

P

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The Rabbit Julia Set with External RaysP(z) = z2 + (−0.12 + 0.78i)

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The Rabbit LaminationP(z) = z2 + (−0.12 + 0.78i)

The rabbit triangle.

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The Rabbit LaminationP(z) = z2 + (−0.12 + 0.78i)

The rabbit lamination

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Hyperbolic lamination pictures courtesy of Clinton Curry

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Laminations of the Unit Disk

Definition

A lamination L is a collections of chords of D, which we callleaves, with the property that any two leaves meet, if at all,in a point of ∂D, andsuch that L has the property that

L∗ := ∂D ∪ {∪L}

is a closed subset of D.We allow degenerate leaves – points of ∂D.

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Extending σd to Leaves

If ` ∈ L is a leaf, we write ` = ab, where a and b are theendpoints of ` in ∂D.We let σd(`) be the chord σd(a)σd(b).If it happens that σd(a) = σd(b), then σd(`) is a point,called a critical value of L, and we say ` is a critical leaf.

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Extending σd to Leaves

If ` ∈ L is a leaf, we write ` = ab, where a and b are theendpoints of ` in ∂D.We let σd(`) be the chord σd(a)σd(b).If it happens that σd(a) = σd(b), then σd(`) is a point,called a critical value of L, and we say ` is a critical leaf.

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Parameterization of Quadratic Polynomials

The general quadratic polynomial function isz 7→ az2 + bz + d .The general quadratic can be affinely conjugated toz 7→ z2 + c, preserving all dynamics.Think of this as a change of coordinates.Thus, all quadratic polynomials are described dynamicallywith one complex parameter.The z-plane is dynamical space, where Julia sets live.What can be said about the c-plane, parameter space?

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σ2 Binary Coordinates

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σ2 Binary Coordinates and RabbitP(z) = z2 + (−0.12 + 0.78i)

In binary coordinates, σ2 is the “forgetful” shift.

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Major and Minor LeavesP(z) = z2 + (−0.12 + 0.78i)

Major leaf is [001,100]. Major leaf is closest in length to 1/2.Minor leaf [001,010] is its image.

This major leaf has endpoints in just one orbit under σ2.24 / 35



Quadratic Parameter Space – the Mandelbrot Set

The Mandelbrot set is the set of points in the c-plane for whichthe orbit of 0 is bounded under z 7→ z2 + c. Equivalently, forwhich the corresponding Julia set is connected.

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Quadratic Minor Lamination – QML– Combinatorial Mandelbrot Set

Thurston: the collection of all minor leaves is itself a lamination.

01

10

001010

011

100

101110

0001

0010

00110100

0110

0111

1000

1001

10111100

1101

1110

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Quadratic Lavaurs Algorithm −→ Recursive QML

1 On a circle, mark each point of period 2 (that is, 01 and10). Join them by a chord.

2 Now mark each point of period 3:001,010,011,100,101,110.Proceeding from the least, in order of increasing angle, jointhem in pairs.

3 Now mark each point of true period 4:0001,0010,0011,0100,0110,0111,1000,1001,1011,1100,1101,1110.Join them in pairs, in order of increasing angle, but so asnot to cross any pre-existing chords.

4 Repeat previous step, increasing period by 1.

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Periodic Minor with Endpoints in Different Orbits

Minor leaf in the QML is [011,100].Dynamical orbit of minor, major, sibling of major, and “guiding”critical chord.

001010

011

100

101 1100110

0011

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“Airplane” Quadratic Julia Set

Minor leaf in the QML is [011,100].

The corresponding point in the Julia set has two ray orbitslanding on it.

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Largest Baby Mandelbrot Set

The two parameter rays 011 and 100 land at its mouth.30 / 35



QML to Period 5

01

10

001010

011

100

101110

0001

0010

0011

0100

0110

0111

1000

1001

10111100

1101

1110

0001000001

00011

00100

001010011000111

010000100101010

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Combinatorial Mandelbrot Set

01

10

001010

011

100

101110

0001

0010

0011

0100

0110

0111

1000

1001

10111100

1101

1110

0001000001

00011

00100

001010011000111

010000100101010

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Parameterization of Cubic Polynomials

The general cubic polynomial function isz 7→ az3 + bz2 + cz + d

The general cubic can be affinely conjugated to differentconvenient forms, preserving all dynamics. One isz 7→ z3 + αz + β

But, to describe all cubic polynomials dynamically requirestwo complex parameters, no less.Thus, cubic parameter space has real dimension 4, notpicturable!One approach is to consider subcollections of cubicpolynomials that can be represented by a single complexparameter.

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“Slices” of Cubic Polynomials

A cubic polynomial has two critical points, counted withmultiplicity.Set one fixed point at 0, fix the derivative at 0 to be λ:z 7→ λz + az2 + z3

Now fixing |λ| ≤ 1 means there is only one “free” criticalpoint.

Require 0 to be a degree 3 critical point: z 7→ z3 + cRequire 0 to be a degree 2 critical point, but map 0immediately to a second critical point:z 7→ z3 + 3cz2 − 2c

Critical points are 0 and −2c.

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References

Brandon L. Barry.On the simplest lamination of a given identity return triangle.PhD Dissertation, UAB, 2015.

David J. Cosper, Jeffrey K. Houghton, John C. Mayer, LukaMernik, and Joseph W. Olson.Central strips of sibling leaves in laminations of the unit disk.To appear: Topology Proceedings, available now electronically.

Dierk Schleicher.On fibers and local connectivity of Mandelbrot and multibrot sets.arXiv:math9902155v1.

William P. Thurston.On the geometry and dynamics of iterated rational maps.Edited by Dierk Schleicher and Nikita Selinger and with anappendix by Schleicher in Complex dynamics. Families andFriends. A K Peters, Ltd., Wellesley, MA, 2009.

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the lavaurs algorithm for thurston's quadratic minor

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