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 : S1 → S2 Peano curve (cont + onto).

Peano curves

γ : S1 → S2 Peano curve (cont + onto). Let x ∼ y iff γ(x) = γ(y), x, y ∈ S1. For each z ∈ S2 γ−1(z) is an equivalence class.

Peano curves

γ : S1 → S2 Peano curve (cont + onto). Let x ∼ y iff γ(x) = γ(y), x, y ∈ S1. For each z ∈ S2 γ−1(z) is an equivalence class. Then S2 ≃ S1/ ∼ S2 quotient of S1.

Peano curves

γ : S1 → S2 Peano curve (cont + onto). Let x ∼ y iff γ(x) = γ(y), x, y ∈ S1. For each z ∈ S2 γ−1(z) is an equivalence class. Then S2 ≃ S1/ ∼ S2 quotient of S1. Three instances where S2 is given in this way.

Peano curves

γ : S1 → S2 Peano curve (cont + onto). Let x ∼ y iff γ(x) = γ(y), x, y ∈ S1. For each z ∈ S2 γ−1(z) is an equivalence class. Then S2 ≃ S1/ ∼ S2 quotient of S1. Three instances where S2 is given in this way.

• 1. Hyperbolic geometry: S2 = ∂H3.

Peano curves

γ : S1 → S2 Peano curve (cont + onto). Let x ∼ y iff γ(x) = γ(y), x, y ∈ S1. For each z ∈ S2 γ−1(z) is an equivalence class. Then S2 ≃ S1/ ∼ S2 quotient of S1. Three instances where S2 is given in this way.

• 1. Hyperbolic geometry: S2 = ∂H3.

M3 hyperbolic 3-manifold that fibers over the circle, S2 = ∂∞π1(M3) (Cannon-Thurston ’07).

Peano curves

γ : S1 → S2 Peano curve (cont + onto). Let x ∼ y iff γ(x) = γ(y), x, y ∈ S1. For each z ∈ S2 γ−1(z) is an equivalence class. Then S2 ≃ S1/ ∼ S2 quotient of S1. Three instances where S2 is given in this way.

• 1. Hyperbolic geometry: S2 = ∂H3.

M3 hyperbolic 3-manifold that fibers over the circle, S2 = ∂∞π1(M3) (Cannon-Thurston ’07).

• 2. Complex dynamics: mating of polynomials.

Peano curves

γ : S1 → S2 Peano curve (cont + onto). Let x ∼ y iff γ(x) = γ(y), x, y ∈ S1. For each z ∈ S2 γ−1(z) is an equivalence class. Then S2 ≃ S1/ ∼ S2 quotient of S1. Three instances where S2 is given in this way.

• 1. Hyperbolic geometry: S2 = ∂H3.

M3 hyperbolic 3-manifold that fibers over the circle, S2 = ∂∞π1(M3) (Cannon-Thurston ’07).

• 2. Complex dynamics: mating of polynomials. Glue two Julia

sets J1, J2 together along their boundaries, S2 = J1⊥ ⊥J2 (Douady-Hubbard ’81).

Peano curves

γ : S1 → S2 Peano curve (cont + onto). Let x ∼ y iff γ(x) = γ(y), x, y ∈ S1. For each z ∈ S2 γ−1(z) is an equivalence class. Then S2 ≃ S1/ ∼ S2 quotient of S1. Three instances where S2 is given in this way.

• 1. Hyperbolic geometry: S2 = ∂H3.

M3 hyperbolic 3-manifold that fibers over the circle, S2 = ∂∞π1(M3) (Cannon-Thurston ’07).

• 2. Complex dynamics: mating of polynomials. Glue two Julia

sets J1, J2 together along their boundaries, S2 = J1⊥ ⊥J2 (Douady-Hubbard ’81).

• 3. Probability: Brownian map,

Peano curves

γ : S1 → S2 Peano curve (cont + onto). Let x ∼ y iff γ(x) = γ(y), x, y ∈ S1. For each z ∈ S2 γ−1(z) is an equivalence class. Then S2 ≃ S1/ ∼ S2 quotient of S1. Three instances where S2 is given in this way.

• 1. Hyperbolic geometry: S2 = ∂H3.

M3 hyperbolic 3-manifold that fibers over the circle, S2 = ∂∞π1(M3) (Cannon-Thurston ’07).

• 2. Complex dynamics: mating of polynomials. Glue two Julia

sets J1, J2 together along their boundaries, S2 = J1⊥ ⊥J2 (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 S2

Maps on S2

Graph embedded in S2 (up to isotopy) = map on S2.

Maps on S2

Graph embedded in S2 (up to isotopy) = map on S2. In general: allow multiple edges, loops.

Maps on S2

Graph embedded in S2 (up to isotopy) = map on S2. In general: allow multiple edges, loops. Often: more restrictive, triangulation, quadrangulations, bound on degrees...

Maps on S2

Graph embedded in S2 (up to isotopy) = map on S2. In general: allow multiple edges, loops. Often: more restrictive, triangulation, quadrangulations, bound on degrees... Equip Graph/map with graph metric.

Maps on S2

Graph embedded in S2 (up to isotopy) = map on S2. 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 S2.

Maps, trees and Peano curves

Consider map on S2. Take spanning tree.

Maps, trees and Peano curves

Consider map on S2. Take spanning tree. Induces dual tree.

Maps, trees and Peano curves

Consider map on S2. 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∗ V = v F = v∗

Detour

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

Mating of polynomials

J1, J2 Julia sets of monic polynomials P1, P2 of degree d ≥ 2.

Mating of polynomials

J1, J2 Julia sets of monic polynomials P1, P2 of degree d ≥ 2. J1, J2 are compact.

Mating of polynomials

J1, J2 Julia sets of monic polynomials P1, P2 of degree d ≥ 2. J1, J2 are compact. Furthermore assume J1, J2 are connected, locally connected, J1, J2 are dendrites (trees).

Mating of polynomials

J1, J2 Julia sets of monic polynomials P1, P2 of degree d ≥ 2. J1, J2 are compact. Furthermore assume J1, J2 are connected, locally connected, J1, J2 are dendrites (trees). Riemann maps C \ D → C \ Jj extend continuously to surjective maps σ1 : S1 → J1 σ2 : S1 → J2. Carath´ eodory loops.

Mating of polynomials

J1, J2 Julia sets of monic polynomials P1, P2 of degree d ≥ 2. J1, J2 are compact. Furthermore assume J1, J2 are connected, locally connected, J1, J2 are dendrites (trees). Riemann maps C \ D → C \ Jj extend continuously to surjective maps σ1 : S1 → J1 σ2 : S1 → J2. Carath´ eodory loops. Consider equivalence relation ∼ on J1 ⊔ J2 generated by σ1, σ2 σ1(z) ∼ σ2(¯ z) for all z ∈ S1 = ∂D.

Mating of polynomials

J1⊥ ⊥J2 := J1 ⊔ J2/ ∼ Mating of J1, J2.

Mating of polynomials

J1⊥ ⊥J2 := J1 ⊔ J2/ ∼ Mating of J1, J2. Dynamics descends to J1⊥ ⊥J2. Carath´ eodory loop is a semi-conjugacy (B¨

• ttcher’s theorem)

S1 z→zd

σ1

• S1

σ1

• J1

P1

J1.

Mating of polynomials

J1⊥ ⊥J2 := J1 ⊔ J2/ ∼ Mating of J1, J2. Dynamics descends to J1⊥ ⊥J2. Carath´ eodory loop is a semi-conjugacy (B¨

• ttcher’s theorem)

S1 z→zd

σ1

• S1

σ1

• J1

P1

J1.

This implies that P1, P2 descend to quotient, i.e., there is a map (mating of P1, P2) P1⊥ ⊥P2 : J1⊥ ⊥J2 → J1⊥ ⊥J2

Peano curves and mating

Semi-conjugacies σ1, σ2 descend to J1⊥ ⊥J2, i.e., there is a map γ : S1 → J1⊥ ⊥J2 such that S1

z→zd

• γ
• S1

γ

• J1⊥

⊥J2 P1 ⊥

⊥P2

J1⊥

⊥J2.

Mating of polynomials

No reason that J1⊥ ⊥J2 is “nice”.

Mating of polynomials

No reason that J1⊥ ⊥J2 is “nice”. Not known if Hausdorff.

Mating of polynomials

No reason that J1⊥ ⊥J2 is “nice”. Not known if Hausdorff. But it is “often” S2. Furthermore P1⊥ ⊥P2 is often (topologically conjugate to) a rational map.

Mating of polynomials

No reason that J1⊥ ⊥J2 is “nice”. Not known if Hausdorff. But it is “often” S2. Furthermore P1⊥ ⊥P2 is often (topologically conjugate to) a rational map.

Theorem (Rees-Shishikura-Tan 1992, 2000)

P1 = z2 + c1, P2 = z2 + c2 postcritically finite (0 has finite orbit). Then P1⊥ ⊥P2 is (topologically conjugate to) a rational map iff c1, c2 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 : S1 → C (onto) such that S1 z→zd

γ

• S1

γ

• C

F

C.

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). Σ × [0, 1] φ

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). Σ × [0, 1] φ M hyperbolic, compact. π1(Σ) ⊳ π1(M) π1(M) = π1(Σ) ⋊ Z

Group invariant Peano curves (Cannon-Thurston)

ι: π1(Σ) → π1(M) homomorphism. π1(Σ) quasi-isometric to H2. π1(M) quasi-isometric to H3.

Group invariant Peano curves (Cannon-Thurston)

ι: π1(Σ) → π1(M) homomorphism. π1(Σ) quasi-isometric to H2. π1(M) quasi-isometric to H3. f : X → Y Quasi-isometry: 1 C |x − y| − A ≤ |f (x) − f (y)| ≤ C|x − y| + A every y ∈ Y has distance at most A from f (X), C, A > 0 constants.

Group invariant Peano curves (Cannon-Thurston)

ι: π1(Σ) → π1(M) homomorphism. π1(Σ) quasi-isometric to H2. π1(M) quasi-isometric to H3. Boundaries at infinity ∂∞π1(Σ) = S1 ∂∞π1(M) = S2. ∂∞ι(π1(Σ)) = S2.

Group invariant Peano curves (Cannon-Thurston)

ι: π1(Σ) → π1(M) homomorphism. π1(Σ) quasi-isometric to H2. π1(M) quasi-isometric to H3. Boundaries at infinity ∂∞π1(Σ) = S1 ∂∞π1(M) = S2. ∂∞ι(π1(Σ)) = S2. So there exists Peano curve γ : S1 → S2 such that for all g ∈ π1(Σ) S1

g

• γ
• S1

γ

• S2

ι(g)

S2.

Brownian map

Consider a map Qn on S2 that is a quadrangulation, i.e., each face has 4 edges/vertices, with n faces.

Brownian map

Consider a map Qn on S2 that is a quadrangulation, i.e., each face has 4 edges/vertices, with n faces. Furthermore the quadrangulation is rooted, i.e., root vertex and root edge (containing the root vertex) are distinguished. Qn set of all such quadrangulations (finite set).

Brownian map

Consider a map Qn on S2 that is a quadrangulation, i.e., each face has 4 edges/vertices, with n faces. Furthermore the quadrangulation is rooted, i.e., root vertex and root edge (containing the root vertex) are distinguished. Qn set of all such quadrangulations (finite set). Take uniform measure on Qn. Random discrete metric space.

Brownian map

Consider a map Qn on S2 that is a quadrangulation, i.e., each face has 4 edges/vertices, with n faces. Furthermore the quadrangulation is rooted, i.e., root vertex and root edge (containing the root vertex) are distinguished. Qn set of all such quadrangulations (finite set). Take uniform measure on Qn. Random discrete metric space.

Technically: prob. meas on (K, dGH) isometry classes of compact metric spaces equipped with Gromov-Hausdorff metric. Separable complete metric space.

Scaling limit

Typically diam Qn ≈ n1/4 (Chassaing-Schaeffer 2004).

Scaling limit

Typically diam Qn ≈ n1/4 (Chassaing-Schaeffer 2004).

Theorem (Le Gall, Miermont 2013)

There exists a random metric space (S, D) such that (Qn, n−1/4dQn) d − → (S, D) as n → ∞. (S, D) is called the Brownian map.

Scaling limit

Typically diam Qn ≈ n1/4 (Chassaing-Schaeffer 2004).

Theorem (Le Gall, Miermont 2013)

There exists a random metric space (S, D) such that (Qn, n−1/4dQn) d − → (S, D) as n → ∞. (S, D) is called the Brownian map. This is conjectured to be a universal object, i.e., (S, D) should be the scaling limit of any “reasonable” map on S2.

Scaling limit

Typically diam Qn ≈ n1/4 (Chassaing-Schaeffer 2004).

Theorem (Le Gall, Miermont 2013)

There exists a random metric space (S, D) such that (Qn, n−1/4dQn) d − → (S, D) as n → ∞. (S, D) is called the Brownian map. This is conjectured to be a universal object, i.e., (S, D) should be the scaling limit of any “reasonable” map on S2. Known: true for triangulations, p-angulations where p ≥ 4 is even (Le Gall). Answers question by Schramm (2006).

Scaling limit

Typically diam Qn ≈ n1/4 (Chassaing-Schaeffer 2004).

Theorem (Le Gall, Miermont 2013)

There exists a random metric space (S, D) such that (Qn, n−1/4dQn) d − → (S, D) as n → ∞. (S, D) is called the Brownian map. This is conjectured to be a universal object, i.e., (S, D) should be the scaling limit of any “reasonable” map on S2. Known: true for triangulations, p-angulations where p ≥ 4 is even (Le Gall). Answers question by Schramm (2006). Known: (S, D) is S2 dimH(S, D) = 4 (Le Gall-Paulin 2008).

Schaeffer bijection

Trees are easier than maps, can enumerate them, construct them inductively.

Schaeffer bijection

Trees are easier than maps, can enumerate them, construct them inductively. Tn set of planar rooted trees with n edges well-labeled, i.e., each vertex v is labeled L(v) ∈ N, s.t. L(∅) = 1 L(v) − L(v′) ∈ {−1, 0, 1}

Schaeffer bijection

Trees are easier than maps, can enumerate them, construct them inductively. Tn set of planar rooted trees with n edges well-labeled, i.e., each vertex v is labeled L(v) ∈ N, s.t. L(∅) = 1 L(v) − L(v′) ∈ {−1, 0, 1}

Theorem (Schaeffer 1998)

There is a bijection Tn → Qn.

Schaeffer bijection

Trees are easier than maps, can enumerate them, construct them inductively. Tn set of planar rooted trees with n edges well-labeled, i.e., each vertex v is labeled L(v) ∈ N, s.t. L(∅) = 1 L(v) − L(v′) ∈ {−1, 0, 1}

Theorem (Schaeffer 1998)

There is a bijection Tn → Qn. Add another vertex “root of root”.

Schaeffer bijection

Trees are easier than maps, can enumerate them, construct them inductively. Tn set of planar rooted trees with n edges well-labeled, i.e., each vertex v is labeled L(v) ∈ N, s.t. L(∅) = 1 L(v) − L(v′) ∈ {−1, 0, 1}

Theorem (Schaeffer 1998)

There is a bijection Tn → Qn. Add another vertex “root of root”. Follow contour (go around the tree).

Schaeffer bijection

Trees are easier than maps, can enumerate them, construct them inductively. Tn set of planar rooted trees with n edges well-labeled, i.e., each vertex v is labeled L(v) ∈ N, s.t. L(∅) = 1 L(v) − L(v′) ∈ {−1, 0, 1}

Theorem (Schaeffer 1998)

There is a bijection Tn → Qn. Add another vertex “root of root”. Follow contour (go around the tree). Connect each vertex to last visited with smaller index.

Continuous random tree

Take uniform measure on Tn. Have diam(Tn) ≈ n1/2.

Continuous random tree

Take uniform measure on Tn. Have diam(Tn) ≈ n1/2.

Theorem (Aldous 1993)

There exists random tree (T, DT) such that (Tn, n−1/2d) d − → (T, DT). continuous random tree CRT. (T, DT) is a geodesic metric tree.

Continuous random tree

Take uniform measure on Tn. Have diam(Tn) ≈ n1/2.

Theorem (Aldous 1993)

There exists random tree (T, DT) such that (Tn, n−1/2d) d − → (T, DT). continuous random tree CRT. (T, DT) is a geodesic metric tree. What is the scaling limit of the labels on the tree?

Continuous random tree

Take uniform measure on Tn. Have diam(Tn) ≈ n1/2.

Theorem (Aldous 1993)

There exists random tree (T, DT) such that (Tn, n−1/2d) d − → (T, DT). continuous random tree CRT. (T, DT) is a geodesic metric tree. What is the scaling limit of the labels on the tree? Along a path in Tn, labels form a random walk.

Brownian labels on CRT

Assign Brownian (Zt)t∈T labels on CRT. Roughly speaking along each (geodesic) path in T Zt is a standard Brownian motion.

Brownian labels on CRT

Assign Brownian (Zt)t∈T labels on CRT. Roughly speaking along each (geodesic) path in T Zt is a standard Brownian motion. Formally Z centered Gaussian process on T, s.t. Z∅ = 0 E[(Zs − Zt)2] = DT(s, t).

Brownian labels on CRT

Assign Brownian (Zt)t∈T labels on CRT. Roughly speaking along each (geodesic) path in T Zt is a standard Brownian motion. Formally Z centered Gaussian process on T, s.t. Z∅ = 0 E[(Zs − Zt)2] = DT(s, t). From the labels one obtains an equivalence relation ≈ on T. s ≈ t :⇔ Zs = Zt = min

u∈[s,t] Zu.

Brownian labels on CRT

Assign Brownian (Zt)t∈T labels on CRT. Roughly speaking along each (geodesic) path in T Zt is a standard Brownian motion. Formally Z centered Gaussian process on T, s.t. Z∅ = 0 E[(Zs − Zt)2] = DT(s, t). From the labels one obtains an equivalence relation ≈ on T. s ≈ t :⇔ Zs = Zt = min

u∈[s,t] Zu.

Have T/≈ ≃ S2, this shows that Brownian map is S2. Can construct metric on the Brownian map from this description.

Brownian map

Is the Brownian map a quasisphere (qs equivalent to S2)?

Brownian map

Is the Brownian map a quasisphere (qs equivalent to S2)? No: not doubling, not bounded turning.

Brownian map

Is the Brownian map a quasisphere (qs equivalent to S2)? No: not doubling, not bounded turning. Everything bad that is possible will happen, but with low probability.

Brownian map

Is the Brownian map a quasisphere (qs equivalent to S2)? No: not doubling, not bounded turning. Everything bad that is possible will happen, but with low probability. What is the right generalization of quasisymmetry?