Laminations of the Unit Disk and Cubic Julia Sets John C. Mayer - PowerPoint PPT Presentation

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Laminations of the Unit Disk and Cubic Julia Sets John C. Mayer Department of Mathematics University of Alabama at Birmingham TOPOSYM 2016, Prague, CZ July 25-29, 2016 1 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set z �→ z 2 − 0 . 12 + 0 . 78 i The Douady Rabbit Julia sets by FractalStream 2 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set The Rabbit Lamination Hyperbolic lamination pictures courtesy of Clinton Curry and Logan Hoehn 3 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Rabbit Juilia Set and Rabbit Lamination Family resemblance? 4 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Outline From Julia Set to Lamination 1 Pullback Laminations 2 Quadratic Cubic Identity Return Triangle From Lamination to Julia Set 3 5 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Outline From Julia Set to Lamination 1 Pullback Laminations 2 Quadratic Cubic Identity Return Triangle From Lamination to Julia Set 3 6 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Outline From Julia Set to Lamination 1 Pullback Laminations 2 Quadratic Cubic Identity Return Triangle From Lamination to Julia Set 3 7 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Julia and Fatou Sets of Polynomials Definitions: Basin of attraction of infinity: B ∞ := { z ∈ C | P n ( z ) → ∞} . Filled Julia set: K ( P ) := C \ B ∞ . Julia set: J ( P ) := boundary of B ∞ = boundary of K ( P ) . Fatou set: F ( P ) := C ∞ \ J ( P ) . Theorems (Facts): J ( P ) is nonempty, compact, and perfect. K ( P ) is full (does not separate C ). Attracting orbits are in Fatou set. Repelling orbits are in Julia set. Examples : P ( z ) = z 2 ; P ( z ) = z d , d > 2; P ( z ) = z 2 − 1, etc. Assume : J ( P ) is connected. 8 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Julia and Fatou Sets of Polynomials Definitions: Basin of attraction of infinity: B ∞ := { z ∈ C | P n ( z ) → ∞} . Filled Julia set: K ( P ) := C \ B ∞ . Julia set: J ( P ) := boundary of B ∞ = boundary of K ( P ) . Fatou set: F ( P ) := C ∞ \ J ( P ) . Theorems (Facts): J ( P ) is nonempty, compact, and perfect. K ( P ) is full (does not separate C ). Attracting orbits are in Fatou set. Repelling orbits are in Julia set. Examples : P ( z ) = z 2 ; P ( z ) = z d , d > 2; P ( z ) = z 2 − 1, etc. Assume : J ( P ) is connected. 9 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Julia and Fatou Sets of Polynomials Definitions: Basin of attraction of infinity: B ∞ := { z ∈ C | P n ( z ) → ∞} . Filled Julia set: K ( P ) := C \ B ∞ . Julia set: J ( P ) := boundary of B ∞ = boundary of K ( P ) . Fatou set: F ( P ) := C ∞ \ J ( P ) . Theorems (Facts): J ( P ) is nonempty, compact, and perfect. K ( P ) is full (does not separate C ). Attracting orbits are in Fatou set. Repelling orbits are in Julia set. Examples : P ( z ) = z 2 ; P ( z ) = z d , d > 2; P ( z ) = z 2 − 1, etc. Assume : J ( P ) is connected. 10 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set The Rabbit Juilia Set and Rabbit Triangle External Rays Landing Angles 2/7 1/7 4/7 11 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set The Rabbit Juilia Set and Rabbit Lamination Down the rabbit hole! 2/7 1/7 4/7 12 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set B¨ ottkher’s Theorem By D ∞ , “the disk at infinity,” we mean C ∞ \ D , the complement of the closed unit disk. Theorem (B¨ ottcher) Let P be a polynomial of degree d. If the filled Julia set K is connected, then there is a conformal isomorphism φ : D ∞ → B ∞ , tangent to the identity at ∞ , that conjugates P to z → z d . 13 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set z �→ z d D ∞ D ∞ ✲ φ φ ❄ ❄ B ∞ B ∞ ✲ P 14 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set z �→ z 2 − 1 Basillica 15 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set z �→ z 2 − 0 . 28136 + 0 . 5326 i Dragon 16 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set z �→ z 2 − 1 . 75 Airplane 17 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Airplane and B-17 Yankee Lady 1 18 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set z �→ z 3 + 0 . 545 + 0 . 539 i Cubic Rabbit 19 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set z �→ z 3 − 0 . 2634 − 1 . 2594 i Helicopter 20 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set √ z �→ z 3 + 2 2 i z 2 Cubic Bug 21 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Cubic Simple Type 1 IRT z �→ z 3 + 3 fz 2 + g f = − 0 . 167026 + 0 . 0384441 i and g = − 0 . 0916222 − 1 . 2734 i 22 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Comparison z �→ z 3 + c z �→ z 3 + 3 fz 2 + g f = − 0 . 167026 + 0 . 0384441 i and g = − 0 . 0916222 − 1 . 2734 i c = − 0 . 2634 − 1 . 2594 i 23 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Laminations of the Disk Laminations were introduced by William Thurston as a way of encoding connected polynomial Julia sets. 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 . 24 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set ?Lamination to Julia Set? The Beginning: Dynamics on the Circle Consider special case P ( z ) = z d on the unit circle ∂ D . z = re 2 π t �→ r d e 2 π ( dt ) . Angle 2 π t �→ 2 π ( dt ) . Measure angles in revolutions: then t �→ dt ( mod 1 ) on ∂ D . Points on ∂ D are coordinatized by [ 0 , 1 ) . 25 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set ?Lamination to Julia Set? The Beginning: Dynamics on the Circle Consider special case P ( z ) = z d on the unit circle ∂ D . z = re 2 π t �→ r d e 2 π ( dt ) . Angle 2 π t �→ 2 π ( dt ) . Measure angles in revolutions: then t �→ dt ( mod 1 ) on ∂ D . Points on ∂ D are coordinatized by [ 0 , 1 ) . 26 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set ?Lamination to Julia Set? The Beginning: Dynamics on the Circle Consider special case P ( z ) = z d on the unit circle ∂ D . z = re 2 π t �→ r d e 2 π ( dt ) . Angle 2 π t �→ 2 π ( dt ) . Measure angles in revolutions: then t �→ dt ( mod 1 ) on ∂ D . Points on ∂ D are coordinatized by [ 0 , 1 ) . 27 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set ?Lamination to Julia Set? The Beginning: Dynamics on the Circle Consider special case P ( z ) = z d on the unit circle ∂ D . z = re 2 π t �→ r d e 2 π ( dt ) . Angle 2 π t �→ 2 π ( dt ) . Measure angles in revolutions: then t �→ dt ( mod 1 ) on ∂ D . Points on ∂ D are coordinatized by [ 0 , 1 ) . 28 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set σ d Dynamics on the Circle σ 2 : t �→ 2 t ( mod 1 ) , angle-doubling. 29 / 76

From Julia Set to Lamination Pullback Laminations From Lamination to Julia Set Induced map σ d on Laminations If ℓ ∈ L is a leaf, we write ℓ = ab , where a and b are the endpoints of ℓ in ∂ D . We let σ d ( ℓ ) be the chord σ d ( a ) σ d ( b ) . 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. 30 / 76

From Julia Set to Lamination Quadratic Pullback Laminations Cubic From Lamination to Julia Set Identity Return Triangle Sibling Invariant Laminations Definition (Sibling Invariant Lamination) A lamination L is said to be sibling d-invariant (or simply invariant if no confusion will result) 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 { 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. 31 / 76