The 2-sphere as a quotient of the circle Daniel Meyer Jacobs - PowerPoint PPT Presentation

The 2-sphere as a quotient of the circle Daniel Meyer Jacobs University April 11, 2014

Peano curves γ : S 1 → S 2 Peano curve (cont + onto).

Peano curves γ : S 1 → S 2 Peano curve (cont + onto). Let x ∼ y iff γ ( x ) = γ ( y ), x , y ∈ S 1 . For each z ∈ S 2 γ − 1 ( z ) is an equivalence class.

Peano curves γ : S 1 → S 2 Peano curve (cont + onto). Let x ∼ y iff γ ( x ) = γ ( y ), x , y ∈ S 1 . For each z ∈ S 2 γ − 1 ( z ) is an equivalence class. Then S 2 ≃ S 1 / ∼ S 2 quotient of S 1 .

Peano curves γ : S 1 → S 2 Peano curve (cont + onto). Let x ∼ y iff γ ( x ) = γ ( y ), x , y ∈ S 1 . For each z ∈ S 2 γ − 1 ( z ) is an equivalence class. Then S 2 ≃ S 1 / ∼ S 2 quotient of S 1 . Three instances where S 2 is given in this way.

Peano curves γ : S 1 → S 2 Peano curve (cont + onto). Let x ∼ y iff γ ( x ) = γ ( y ), x , y ∈ S 1 . For each z ∈ S 2 γ − 1 ( z ) is an equivalence class. Then S 2 ≃ S 1 / ∼ S 2 quotient of S 1 . Three instances where S 2 is given in this way. 1. Hyperbolic geometry: S 2 = ∂ H 3 .

Peano curves γ : S 1 → S 2 Peano curve (cont + onto). Let x ∼ y iff γ ( x ) = γ ( y ), x , y ∈ S 1 . For each z ∈ S 2 γ − 1 ( z ) is an equivalence class. Then S 2 ≃ S 1 / ∼ S 2 quotient of S 1 . Three instances where S 2 is given in this way. 1. Hyperbolic geometry: S 2 = ∂ H 3 . M 3 hyperbolic 3-manifold that fibers over the circle, S 2 = ∂ ∞ π 1 ( M 3 ) (Cannon-Thurston ’07).

Peano curves γ : S 1 → S 2 Peano curve (cont + onto). Let x ∼ y iff γ ( x ) = γ ( y ), x , y ∈ S 1 . For each z ∈ S 2 γ − 1 ( z ) is an equivalence class. Then S 2 ≃ S 1 / ∼ S 2 quotient of S 1 . Three instances where S 2 is given in this way. 1. Hyperbolic geometry: S 2 = ∂ H 3 . M 3 hyperbolic 3-manifold that fibers over the circle, S 2 = ∂ ∞ π 1 ( M 3 ) (Cannon-Thurston ’07). 2. Complex dynamics: mating of polynomials.

Peano curves γ : S 1 → S 2 Peano curve (cont + onto). Let x ∼ y iff γ ( x ) = γ ( y ), x , y ∈ S 1 . For each z ∈ S 2 γ − 1 ( z ) is an equivalence class. Then S 2 ≃ S 1 / ∼ S 2 quotient of S 1 . Three instances where S 2 is given in this way. 1. Hyperbolic geometry: S 2 = ∂ H 3 . M 3 hyperbolic 3-manifold that fibers over the circle, S 2 = ∂ ∞ π 1 ( M 3 ) (Cannon-Thurston ’07). 2. Complex dynamics: mating of polynomials. Glue two Julia sets J 1 , J 2 together along their boundaries, S 2 = J 1 ⊥ ⊥ J 2 (Douady-Hubbard ’81).

Peano curves γ : S 1 → S 2 Peano curve (cont + onto). Let x ∼ y iff γ ( x ) = γ ( y ), x , y ∈ S 1 . For each z ∈ S 2 γ − 1 ( z ) is an equivalence class. Then S 2 ≃ S 1 / ∼ S 2 quotient of S 1 . Three instances where S 2 is given in this way. 1. Hyperbolic geometry: S 2 = ∂ H 3 . M 3 hyperbolic 3-manifold that fibers over the circle, S 2 = ∂ ∞ π 1 ( M 3 ) (Cannon-Thurston ’07). 2. Complex dynamics: mating of polynomials. Glue two Julia sets J 1 , J 2 together along their boundaries, S 2 = J 1 ⊥ ⊥ J 2 (Douady-Hubbard ’81). 3. Probability: Brownian map,

Peano curves γ : S 1 → S 2 Peano curve (cont + onto). Let x ∼ y iff γ ( x ) = γ ( y ), x , y ∈ S 1 . For each z ∈ S 2 γ − 1 ( z ) is an equivalence class. Then S 2 ≃ S 1 / ∼ S 2 quotient of S 1 . Three instances where S 2 is given in this way. 1. Hyperbolic geometry: S 2 = ∂ H 3 . M 3 hyperbolic 3-manifold that fibers over the circle, S 2 = ∂ ∞ π 1 ( M 3 ) (Cannon-Thurston ’07). 2. Complex dynamics: mating of polynomials. Glue two Julia sets J 1 , J 2 together along their boundaries, S 2 = J 1 ⊥ ⊥ J 2 (Douady-Hubbard ’81). 3. Probability: Brownian map, random metric sphere, 2-dimensional analog of Brownian motion. Limit of random triangulation (Le Gall ’07).

Maps on S 2

Maps on S 2 Graph embedded in S 2 (up to isotopy) = map on S 2 .

Maps on S 2 Graph embedded in S 2 (up to isotopy) = map on S 2 . In general: allow multiple edges, loops.

Maps on S 2 Graph embedded in S 2 (up to isotopy) = map on S 2 . In general: allow multiple edges, loops. Often: more restrictive, triangulation, quadrangulations, bound on degrees...

Maps on S 2 Graph embedded in S 2 (up to isotopy) = map on S 2 . In general: allow multiple edges, loops. Often: more restrictive, triangulation, quadrangulations, bound on degrees... Equip Graph/map with graph metric.

Maps on S 2 Graph embedded in S 2 (up to isotopy) = map on S 2 . In general: allow multiple edges, loops. Often: more restrictive, triangulation, quadrangulations, bound on degrees... Equip Graph/map with graph metric. Discrete approximation of metric sphere.

Maps, trees and Peano curves Consider map on S 2 .

Maps, trees and Peano curves Consider map on S 2 . Take spanning tree.

Maps, trees and Peano curves Consider map on S 2 . Take spanning tree. Induces dual tree.

Maps, trees and Peano curves Consider map on S 2 . Take spanning tree. Induces dual tree. Between the trees there is a Jordan curve through all faces. Discrete approximation of Peano curve.

Detour For trees have v ∗ = e ∗ + 1 v = e + 1

Detour For trees have v ∗ = e ∗ + 1 v = e + 1 Note E = e + e ∗ F = v ∗ V = v

Detour For trees have v ∗ = e ∗ + 1 v = e + 1 Note E = e + e ∗ F = v ∗ V = v V − E + F = v − e − e ∗ + v ∗ = 2 .

Mating of polynomials J 1 , J 2 Julia sets of monic polynomials P 1 , P 2 of degree d ≥ 2.

Mating of polynomials J 1 , J 2 Julia sets of monic polynomials P 1 , P 2 of degree d ≥ 2. J 1 , J 2 are compact.

Mating of polynomials J 1 , J 2 Julia sets of monic polynomials P 1 , P 2 of degree d ≥ 2. J 1 , J 2 are compact. Furthermore assume J 1 , J 2 are connected, locally connected, J 1 , J 2 are dendrites (trees).

Mating of polynomials J 1 , J 2 Julia sets of monic polynomials P 1 , P 2 of degree d ≥ 2. J 1 , J 2 are compact. Furthermore assume J 1 , J 2 are connected, locally connected, J 1 , J 2 are dendrites (trees). Riemann maps � C \ D → � C \ J j extend continuously to surjective maps σ 1 : S 1 → J 1 σ 2 : S 1 → J 2 . Carath´ eodory loops.

Mating of polynomials J 1 , J 2 Julia sets of monic polynomials P 1 , P 2 of degree d ≥ 2. J 1 , J 2 are compact. Furthermore assume J 1 , J 2 are connected, locally connected, J 1 , J 2 are dendrites (trees). Riemann maps � C \ D → � C \ J j extend continuously to surjective maps σ 1 : S 1 → J 1 σ 2 : S 1 → J 2 . Carath´ eodory loops. Consider equivalence relation ∼ on J 1 ⊔ J 2 generated by σ 1 , σ 2 σ 1 ( z ) ∼ σ 2 (¯ z ) for all z ∈ S 1 = ∂ D .

Mating of polynomials J 1 ⊥ ⊥ J 2 := J 1 ⊔ J 2 / ∼ Mating of J 1 , J 2 .

� � Mating of polynomials J 1 ⊥ ⊥ J 2 := J 1 ⊔ J 2 / ∼ Mating of J 1 , J 2 . Dynamics descends to J 1 ⊥ ⊥ J 2 . Carath´ eodory loop is a semi-conjugacy (B¨ ottcher’s theorem) S 1 z �→ z d � S 1 σ 1 σ 1 � J 1 . J 1 P 1

� � Mating of polynomials J 1 ⊥ ⊥ J 2 := J 1 ⊔ J 2 / ∼ Mating of J 1 , J 2 . Dynamics descends to J 1 ⊥ ⊥ J 2 . Carath´ eodory loop is a semi-conjugacy (B¨ ottcher’s theorem) S 1 z �→ z d � S 1 σ 1 σ 1 � J 1 . J 1 P 1 This implies that P 1 , P 2 descend to quotient, i.e., there is a map (mating of P 1 , P 2 ) P 1 ⊥ ⊥ P 2 : J 1 ⊥ ⊥ J 2 → J 1 ⊥ ⊥ J 2

� � � Peano curves and mating Semi-conjugacies σ 1 , σ 2 descend to J 1 ⊥ ⊥ J 2 , i.e., there is a map γ : S 1 → J 1 ⊥ ⊥ J 2 such that z �→ z d S 1 S 1 γ γ � J 1 ⊥ J 1 ⊥ ⊥ J 2 P 1 ⊥ ⊥ J 2 . ⊥ P 2

Mating of polynomials No reason that J 1 ⊥ ⊥ J 2 is “nice”.

Mating of polynomials No reason that J 1 ⊥ ⊥ J 2 is “nice”. Not known if Hausdorff.

Mating of polynomials No reason that J 1 ⊥ ⊥ J 2 is “nice”. Not known if Hausdorff. But it is “often” S 2 . Furthermore P 1 ⊥ ⊥ P 2 is often (topologically conjugate to) a rational map.

Mating of polynomials No reason that J 1 ⊥ ⊥ J 2 is “nice”. Not known if Hausdorff. But it is “often” S 2 . Furthermore P 1 ⊥ ⊥ P 2 is often (topologically conjugate to) a rational map. Theorem (Rees-Shishikura-Tan 1992, 2000) P 1 = z 2 + c 1 , P 2 = z 2 + c 2 postcritically finite ( 0 has finite orbit). Then P 1 ⊥ ⊥ P 2 is (topologically conjugate to) a rational map iff c 1 , c 2 are not in conjugate limbs of the Mandelbrot set.

� � Unmating of rational maps Given rational map f : � C → � C . Does f arise as a (is topologically conjugate to) mating? Theorem (M) Let f : � C → � C rational postcritically finite, J ( f ) = � C , then every sufficiently high iterate F = f n arises as a mating. Thus there is a Peano curve γ : S 1 → � C (onto) such that S 1 z �→ z d � S 1 γ γ � � � C C . F

Group invariant Peano curves (Cannon-Thurston) Manifold that fibers over the circle: Σ closed hyperbolic 2-manifold, φ : Σ → Σ Pseudo-Anosov . M = Σ × [0 , 1] / ∼ , where ( s , 0) ∼ ( φ ( s ) , 1) . φ Σ × [0 , 1]

Group invariant Peano curves (Cannon-Thurston) Manifold that fibers over the circle: Σ closed hyperbolic 2-manifold, φ : Σ → Σ Pseudo-Anosov . M = Σ × [0 , 1] / ∼ , where ( s , 0) ∼ ( φ ( s ) , 1) . M hyperbolic, compact. φ π 1 (Σ) ⊳ π 1 ( M ) π 1 ( M ) = π 1 (Σ) ⋊ Z Σ × [0 , 1]