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 1 / 87
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 2 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations Outline Polynomial Julia Sets and Laminations 1 Critical Chords and Pullbacks 2 Critical Portraits, Dual Graphs, and Simplest Laminations 3 3 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations Outline Polynomial Julia Sets and Laminations 1 Critical Chords and Pullbacks 2 Critical Portraits, Dual Graphs, and Simplest Laminations 3 4 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations Outline Polynomial Julia Sets and Laminations 1 Critical Chords and Pullbacks 2 Critical Portraits, Dual Graphs, and Simplest Laminations 3 5 / 87
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 ) = a d z d + a d − 1 z d − 1 + · · · + a 1 z + a 0 Compactify C to C ∞ . For P , ∞ is attracting fixed point: for z ∈ C with | z | sufficiently large, lim n →∞ P n ( z ) = ∞ . Basin of attraction of ∞ : n →∞ P n ( z ) = ∞} B ∞ := { z ∈ C | lim B ∞ is an open set. 6 / 87
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). 7 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations z �→ z 2 − 1 Basillica Julia set pictures by Fractalstream 8 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations z �→ z 2 + ( − 0 . 12 + 0 . 78 i ) Douady Rabbit 9 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations z �→ z 2 + ( 0 . 057 + 0 . 713 i ) Twisted Rabbit 10 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations z �→ z 2 − 1 . 75 Airplane 11 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations z �→ z 3 + ( 0 . 545 + 0 . 539 i ) Minnie Mouse 12 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations z �→ z 3 + ( − 0 . 2634 − 1 . 2594 i ) Helicopter 13 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations z �→ z 3 + 3 ( 0 . 785415 i ) z 2 Scorpion/Scepter 14 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations z �→ z 3 + 3 ( − 0 . 5 ) z 2 + ( 0 . 75 + 0 . 661438 i ) Butterfly 15 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations z �→ z 3 + ( 0 . 20257 + 1 . 095 i ) Ninja Throwing Star 16 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations The Simplest Julia Set – the Unit Circle ∂ D re 2 π i t �→ r 2 e 2 π i 2 t P ( z ) = z 2 The complement C ∞ \ D of the closed unit disk is the basin of attraction, B ∞ , of infinity. 17 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations Dynamics on the Unit Circle Consider P ( z ) = z d on the unit circle ∂ D . z = re 2 π t �→ r d e 2 π i ( dt ) − → Angle 2 π t �→ 2 π ( dt ) . Measure angles in revolutions: Points on ∂ D are coordinatized by [ 0 , 1 ) . σ d : t �→ dt ( mod 1 ) on ∂ D . Example d = 2: 18 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations Bottcher’s Theorem z �→ z d D ∞ D ∞ ✲ φ φ ❄ ❄ B ∞ B ∞ ✲ P 19 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations P ( z ) = z 2 + ( − 0 . 12 + 0 . 78 i ) External Rays 2/7 1/7 4/7 20 / 87
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 of encoding connected polynomial Julia sets. Coincident external rays Rabbit triangle 2/7 1/7 4/7 21 / 87
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 2/7 1/7 4/7 Hyperbolic lamination pictures courtesy of Logan Hoehn 22 / 87
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. 23 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations Extending σ d 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 σ d ( ℓ ) to be the chord σ d ( a ) σ d ( b ) . The length of a chord is the length of the shorter arc of the circle subtended. If it happens that σ d ( a ) = σ d ( b ) , then σ d ( ℓ ) is a point, called a critical value of L , and we say ℓ is a critical leaf. 24 / 87
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: (Forward Invariant) For every ℓ ∈ L , σ d ( ℓ ) ∈ L . 1 (Backward Invariant) For every non-degenerate ℓ ′ ∈ L , 2 there is a leaf ℓ ∈ L such that σ d ( ℓ ) = ℓ ′ . (Sibling Invariant) For every ℓ 1 ∈ L with σ d ( ℓ 1 ) = ℓ ′ , a 3 non-degenerate leaf, there is a full sibling collection of pairwise disjoint leaves { ℓ 1 , ℓ 2 , . . . , ℓ d } ⊂ L such that σ d ( ℓ i ) = ℓ ′ . Conditions (1), (2) and (3) allow generating a sibling invariant lamination from a finite amount of initial data. 25 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations σ 2 Binary Coordinates Location dynamically defined. 0 010 001 00 01 0001 000 011 0000 111 100 11 10 101 110 1 26 / 87
Polynomial Julia Sets and Laminations Critical Chords and Pullbacks Critical Portraits, Dual Graphs, and Simplest Laminations σ 2 Binary Coordinates and Rabbit In binary coordinates, σ 2 is the “forgetful” shift. The overline means the coordinates repeat. 27 / 87
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 σ d is a collection of d branches τ 1 , τ 2 , . . . , τ d of the inverse of σ d whose ranges partition ∂ D . 010 001 100 Data : Forward invariant lamination. 28 / 87
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