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 = ∂H3 .
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 = ∂H3 .
M 3 hyperbolic 3-manifold that fibers over the circle,
S 2 = ∂∞ π1 (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 = ∂H3 .
M 3 hyperbolic 3-manifold that fibers over the circle,
S 2 = ∂∞ π1 (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 = ∂H3 .
M 3 hyperbolic 3-manifold that fibers over the circle,
S 2 = ∂∞ π1 (M 3 ) (Cannon-Thurston ’07).
2. Complex dynamics: mating of polynomials. Glue two Julia
sets J1 , J2 together along their boundaries, S 2 = J1 ⊥⊥J2
(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 = ∂H3 .
M 3 hyperbolic 3-manifold that fibers over the circle,
S 2 = ∂∞ π1 (M 3 ) (Cannon-Thurston ’07).
2. Complex dynamics: mating of polynomials. Glue two Julia
sets J1 , J2 together along their boundaries, S 2 = J1 ⊥⊥J2
(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 = ∂H3 .
M 3 hyperbolic 3-manifold that fibers over the circle,
S 2 = ∂∞ π1 (M 3 ) (Cannon-Thurston ’07).
2. Complex dynamics: mating of polynomials. Glue two Julia
sets J1 , J2 together along their boundaries, S 2 = 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 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∗
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).
b \D→C
b \ Jj extend continuously to surjective
Riemann maps C
maps
σ1 : S 1 → J 1
σ2 : S 1 → J2 .
Carathéodory 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).
b \D→C
b \ Jj extend continuously to surjective
Riemann maps C
maps
σ1 : S 1 → J 1
σ2 : S 1 → J2 .
Carathéodory loops.
Consider equivalence relation ∼ on J1 t J2 generated by σ1 , σ2
σ1 (z) ∼ σ2 (z̄)
for all z ∈ S 1 = ∂D.
Mating of polynomials
J1 ⊥⊥J2 := J1 t J2 / ∼
Mating of J1 , J2 .
Mating of polynomials
J1 ⊥⊥J2 := J1 t J2 / ∼
Mating of J1 , J2 .
Dynamics descends to J1 ⊥⊥J2 . Carathéodory loop is a
semi-conjugacy (Böttcher’s theorem)
S1
σ1
z7→z d
J1
/ S1
P1
σ1
/ J1 .
Mating of polynomials
J1 ⊥⊥J2 := J1 t J2 / ∼
Mating of J1 , J2 .
Dynamics descends to J1 ⊥⊥J2 . Carathéodory loop is a
semi-conjugacy (Böttcher’s theorem)
S1
σ1
z7→z d
J1
/ S1
P1
σ1
/ 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 : S 1 → J1 ⊥⊥J2 such that
S1
γ
J1 ⊥⊥J2
z7→z d
/
/ S1
γ
J ⊥⊥J2 .
P1 ⊥
⊥ P2 1
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” S 2 . 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” S 2 . Furthermore P1 ⊥⊥P2 is often (topologically
conjugate to) a rational map.
Theorem (Rees-Shishikura-Tan 1992, 2000)
P1 = z 2 + c1 , P2 = z 2 + 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
b → C.
b Does f arise as a (is topologically
Given rational map f : C
conjugate to) mating?
Theorem (M)
b →C
b rational postcritically finite, J(f ) = C,
b then every
Let f : C
n
sufficiently high iterate F = f arises as a mating.
b (onto) such that
Thus there is a Peano curve γ : S 1 → C
S1
γ
z7→z d /
b
C
S1
F
γ
/b
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).
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
π1 (Σ) / π1 (M)
π1 (M) = π1 (Σ) o Z
Σ × [0, 1]
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
|x − y | − A ≤ |f (x) − f (y )| ≤ C |x − y | + A
C
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) = S 1
∂∞ π1 (M) = S 2 .
∂∞ ι(π1 (Σ)) = S 2 .
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) = S 1
∂∞ π1 (M) = S 2 .
∂∞ ι(π1 (Σ)) = S 2 .
So there exists Peano curve γ : S 1 → S 2 such that for all
g ∈ π1 (Σ)
S1
g
/ S1
γ
γ
S2
ι(g )
/ S 2.
Brownian map
Consider a map Qn on S 2 that is a quadrangulation, i.e., each face
has 4 edges/vertices, with n faces.
Brownian map
Consider a map Qn on S 2 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 S 2 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 S 2 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
d
(Qn , n−1/4 dQn ) −
→ (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
d
(Qn , n−1/4 dQn ) −
→ (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 S 2 .
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
d
(Qn , n−1/4 dQn ) −
→ (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 S 2 .
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
d
(Qn , n−1/4 dQn ) −
→ (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 S 2 .
Known: true for triangulations, p-angulations where p ≥ 4 is even
(Le Gall). Answers question by Schramm (2006).
Known: (S, D) is S 2
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 0 ) ∈ {−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 0 ) ∈ {−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 0 ) ∈ {−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 0 ) ∈ {−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 0 ) ∈ {−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
d
(Tn , n−1/2 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
d
(Tn , n−1/2 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
d
(Tn , n−1/2 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 Zu .
u∈[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 Zu .
u∈[s,t]
Have T / ≈ ' S 2 , this shows that Brownian map is S 2 .
Can construct metric on the Brownian map from this description.
Brownian map
Is the Brownian map a quasisphere (qs equivalent to S 2 )?
Brownian map
Is the Brownian map a quasisphere (qs equivalent to S 2 )?
No: not doubling, not bounded turning.
Brownian map
Is the Brownian map a quasisphere (qs equivalent to S 2 )?
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 S 2 )?
No: not doubling, not bounded turning.
Everything bad that is possible will happen, but with low
probability.
What is the right generalization of quasisymmetry?