Paths in Euclidean space
L. C. Hoehn ([email protected])
joint with:
L. G. Oversteegen
E. D. Tymchatyn
Nipissing University
July 23, 2014
Summer Topology Conference
College of Staten Island
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
1/7
Paths
Let n ≥ 1 be fixed.
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
2/7
Paths
Let n ≥ 1 be fixed.
A path is a continuous function γ : [0, 1] → Rn .
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
2/7
Paths
Let n ≥ 1 be fixed.
A path is a continuous function γ : [0, 1] → Rn .
C[0, 1] = {γ : γ is a path}, with metric
dsup (γ1 , γ2 ) = sup{|γ1 (t) − γ2 (t)| : t ∈ [0, 1]}
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
2/7
Paths
Let n ≥ 1 be fixed.
A path is a continuous function γ : [0, 1] → Rn .
C[0, 1] = {γ : γ is a path}, with metric
dsup (γ1 , γ2 ) = sup{|γ1 (t) − γ2 (t)| : t ∈ [0, 1]}
Reparameterization: γ1 ≈ γ2 if there are non-decreasing onto maps
m1 , m2 : [0, 1] → [0, 1] such that
γi is constant on each fiber mi−1 (s), s ∈ [0, 1], for both i = 1, 2; and
γ1 ◦ m1−1 = γ2 ◦ m2−1 .
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
2/7
Paths
Let n ≥ 1 be fixed.
A path is a continuous function γ : [0, 1] → Rn .
C[0, 1] = {γ : γ is a path}, with metric
dsup (γ1 , γ2 ) = sup{|γ1 (t) − γ2 (t)| : t ∈ [0, 1]}
Reparameterization: γ1 ≈ γ2 if there are non-decreasing onto maps
m1 , m2 : [0, 1] → [0, 1] such that
γi is constant on each fiber mi−1 (s), s ∈ [0, 1], for both i = 1, 2; and
γ1 ◦ m1−1 = γ2 ◦ m2−1 .
Let Π = {[γ] : γ is a path} = C[0, 1] /≈
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
2/7
Paths
Let n ≥ 1 be fixed.
A path is a continuous function γ : [0, 1] → Rn .
C[0, 1] = {γ : γ is a path}, with metric
dsup (γ1 , γ2 ) = sup{|γ1 (t) − γ2 (t)| : t ∈ [0, 1]}
Reparameterization: γ1 ≈ γ2 if there are non-decreasing onto maps
m1 , m2 : [0, 1] → [0, 1] such that
γi is constant on each fiber mi−1 (s), s ∈ [0, 1], for both i = 1, 2; and
γ1 ◦ m1−1 = γ2 ◦ m2−1 .
Let Π = {[γ] : γ is a path} = C[0, 1] /≈
Metric d on Π:
d([γ1 ], [γ2 ]) = inf{dsup (λ1 , λ2 ) : λ1 ∈ [γ1 ], λ2 ∈ [γ2 ]}
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
2/7
Alternative path length (in R2 )
Given a path γ : [0, 1] → R2 . . .
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
3/7
Alternative path length (in R2 )
Given a path γ : [0, 1] → R2 . . .
Cut by family of lines obtained from
R × Z by
Translating vertically by x ∈ [0, 1];
Rotating by angle tπ, t ∈ [0, 1];
Scaling by factor µ ∈ (0, 1].
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
3/7
Alternative path length (in R2 )
Given a path γ : [0, 1] → R2 . . .
Cut by family of lines obtained from
R × Z by
Translating vertically by x ∈ [0, 1];
Rotating by angle tπ, t ∈ [0, 1];
Scaling by factor µ ∈ (0, 1].
This splits path into subpaths Cn .
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
3/7
Alternative path length (in R2 )
Given a path γ : [0, 1] → R2 . . .
Cut by family of lines obtained from
R × Z by
Translating vertically by x ∈ [0, 1];
Rotating by angle tπ, t ∈ [0, 1];
Scaling by factor µ ∈ (0, 1].
This splits path into subpaths Cn .
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
3/7
Alternative path length (in R2 )
Given a path γ : [0, 1] → R2 . . .
Cut by family of lines obtained from
R × Z by
Translating vertically by x ∈ [0, 1];
Rotating by angle tπ, t ∈ [0, 1];
Scaling by factor µ ∈ (0, 1].
This splits path into subpaths Cn .
kγ(Cn )kt = distance the section Cn
goes across its strip.
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
3/7
Alternative path length (in R2 )
Given a path γ : [0, 1] → R2 . . .
Cut by family of lines obtained from
R × Z by
Translating vertically by x ∈ [0, 1];
Rotating by angle tπ, t ∈ [0, 1];
Scaling by factor µ ∈ (0, 1].
This splits path into subpaths Cn .
kγ(Cn )kt = distance the section Cn
goes across its strip.
Lx,t,µ (γ) =
∞
X
kγ(Cn )kt
n=1
2n
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
3/7
Alternative path length (in R2 )
Given a path γ : [0, 1] → R2 . . .
Cut by family of lines obtained from
R × Z by
Translating vertically by x ∈ [0, 1];
Rotating by angle tπ, t ∈ [0, 1];
Scaling by factor µ ∈ (0, 1].
This splits path into subpaths Cn .
kγ(Cn )kt = distance the section Cn
goes across its strip.
Lx,t,µ (γ) =
∞
X
kγ(Cn )kt
n=1
2n
Z 1Z 1Z
1
Lx,t,µ (γ) dx dt dµ.
len(γ) =
0
0
0
Compare with Cannon, Conner, Zastrow (2002) and Morse (1936).
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
3/7
Properties of len
For any path γ : [0, 1] → Rn :
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
4/7
Properties of len
For any path γ : [0, 1] → Rn :
0 ≤ len(γ) < 1
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
4/7
Properties of len
For any path γ : [0, 1] → Rn :
0 ≤ len(γ) < 1
If Φ : Rn → Rn is an isometry, then len(Φ ◦ γ) = len(γ)
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
4/7
Properties of len
For any path γ : [0, 1] → Rn :
0 ≤ len(γ) < 1
If Φ : Rn → Rn is an isometry, then len(Φ ◦ γ) = len(γ)
If λ ≈ γ, then len(λ) = len(γ)
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
4/7
Properties of len
For any path γ : [0, 1] → Rn :
0 ≤ len(γ) < 1
If Φ : Rn → Rn is an isometry, then len(Φ ◦ γ) = len(γ)
If λ ≈ γ, then len(λ) = len(γ)
Let S parameterize the straight line segment from γ(0) to γ(1). Then
len(S) ≤ len(γ)
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
4/7
Properties of len
For any path γ : [0, 1] → Rn :
0 ≤ len(γ) < 1
If Φ : Rn → Rn is an isometry, then len(Φ ◦ γ) = len(γ)
If λ ≈ γ, then len(λ) = len(γ)
Let S parameterize the straight line segment from γ(0) to γ(1). Then
len(S) ≤ len(γ)
If [a, b] ⊆ [0, 1], then len(γ[a,b] ) ≤ len(γ)
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
4/7
Properties of len
For any path γ : [0, 1] → Rn :
0 ≤ len(γ) < 1
If Φ : Rn → Rn is an isometry, then len(Φ ◦ γ) = len(γ)
If λ ≈ γ, then len(λ) = len(γ)
Let S parameterize the straight line segment from γ(0) to γ(1). Then
len(S) ≤ len(γ)
If [a, b] ⊆ [0, 1], then len(γ[a,b] ) ≤ len(γ)
If c ∈ (0, 1), then len(γ) ≤ len(γ[0,c] ) + len(γ[c,1] )
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
4/7
Properties of len
For any path γ : [0, 1] → Rn :
0 ≤ len(γ) < 1
If Φ : Rn → Rn is an isometry, then len(Φ ◦ γ) = len(γ)
If λ ≈ γ, then len(λ) = len(γ)
Let S parameterize the straight line segment from γ(0) to γ(1). Then
len(S) ≤ len(γ)
If [a, b] ⊆ [0, 1], then len(γ[a,b] ) ≤ len(γ)
If c ∈ (0, 1), then len(γ) ≤ len(γ[0,c] ) + len(γ[c,1] )
len is a continuous function C[0, 1] → R
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
4/7
Properties of len
For any path γ : [0, 1] → Rn :
0 ≤ len(γ) < 1
If Φ : Rn → Rn is an isometry, then len(Φ ◦ γ) = len(γ)
If λ ≈ γ, then len(λ) = len(γ)
Let S parameterize the straight line segment from γ(0) to γ(1). Then
len(S) ≤ len(γ)
If [a, b] ⊆ [0, 1], then len(γ[a,b] ) ≤ len(γ)
If c ∈ (0, 1), then len(γ) ≤ len(γ[0,c] ) + len(γ[c,1] )
len is a continuous function C[0, 1] → R
The function len can be defined analogously for any map γ from a locally
connected continuum X to Rn .
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
4/7
Parameterization by len
Given a path γ, the parameterization of γ by len is the path γ
e ∈ [γ] such
that len(e
γ [0,s] ) = s · len(γ) for all s ∈ [0, 1].
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
5/7
Parameterization by len
Given a path γ, the parameterization of γ by len is the path γ
e ∈ [γ] such
that len(e
γ [0,s] ) = s · len(γ) for all s ∈ [0, 1].
e = the set of all standard parameterizations of paths [0, 1] → Rn .
Π
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
5/7
Parameterization by len
Given a path γ, the parameterization of γ by len is the path γ
e ∈ [γ] such
that len(e
γ [0,s] ) = s · len(γ) for all s ∈ [0, 1].
e = the set of all standard parameterizations of paths [0, 1] → Rn .
Π
Theorem
e is a closed subset of C[0, 1], and the function [γ] 7→ γ
Π
e is a
e
homeomorphism from Π to Π.
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
5/7
Parameterization by len
Given a path γ, the parameterization of γ by len is the path γ
e ∈ [γ] such
that len(e
γ [0,s] ) = s · len(γ) for all s ∈ [0, 1].
e = the set of all standard parameterizations of paths [0, 1] → Rn .
Π
Theorem
e is a closed subset of C[0, 1], and the function [γ] 7→ γ
Π
e is a
e
homeomorphism from Π to Π.
Corollary
A set F ⊆ Π is closed (respectively, compact) if and only if
Fe = {e
γ : [γ] ∈ F} is a closed (respectively, compact) subset of C[0, 1].
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
5/7
Families of paths
Let F ⊆ Π.
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
6/7
Families of paths
Let F ⊆ Π.
(†): For each ε > 0, there is a positive integer N such that for
every [γ] ∈ F, there is no collection of more than N pairwise
disjoint subintervals of [0, 1] whose images under γ have
diameters ≥ ε.
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
6/7
Families of paths
Let F ⊆ Π.
(†): For each ε > 0, there is a positive integer N such that for
every [γ] ∈ F, there is no collection of more than N pairwise
disjoint subintervals of [0, 1] whose images under γ have
diameters ≥ ε.
Theorem
If (†) holds, then the family Fe = {e
γ : [γ] ∈ F} is equicontinuous.
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
6/7
Families of paths
Let F ⊆ Π.
(†): For each ε > 0, there is a positive integer N such that for
every [γ] ∈ F, there is no collection of more than N pairwise
disjoint subintervals of [0, 1] whose images under γ have
diameters ≥ ε.
Theorem
If (†) holds, then the family Fe = {e
γ : [γ] ∈ F} is equicontinuous.
Conversely, if an equicontinuous family can be formed by choosing
parameterizations of all the paths in F, then (†) holds.
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
6/7
Families of paths
Let F ⊆ Π.
(†): For each ε > 0, there is a positive integer N such that for
every [γ] ∈ F, there is no collection of more than N pairwise
disjoint subintervals of [0, 1] whose images under γ have
diameters ≥ ε.
Theorem
If (†) holds, then the family Fe = {e
γ : [γ] ∈ F} is equicontinuous.
Conversely, if an equicontinuous family can be formed by choosing
parameterizations of all the paths in F, then (†) holds.
Theorem
F is compact if and only if:
1
the set {γ(0) : [γ] ∈ F} is bounded; and
2
F satisfies the property (†).
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
6/7
Midpoint parameterization
Given a path γ, let m ∈ (0, 1) be such that len(γ[0,m] ) = len(γ[m,1] ).
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
7/7
Midpoint parameterization
Given a path γ, let m ∈ (0, 1) be such that len(γ[0,m] ) = len(γ[m,1] ).
The midpoint parameterization of γ is the path γ ∗ ∈ [γ] such that
γ ∗ [ 1 ,0] is the parameterization by len of γ[m,0] ; and
2
γ ∗ [ 1 ,1] is the parameterization by len of γ[m,1] .
2
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
7/7
Midpoint parameterization
Given a path γ, let m ∈ (0, 1) be such that len(γ[0,m] ) = len(γ[m,1] ).
The midpoint parameterization of γ is the path γ ∗ ∈ [γ] such that
γ ∗ [ 1 ,0] is the parameterization by len of γ[m,0] ; and
2
γ ∗ [ 1 ,1] is the parameterization by len of γ[m,1] .
2
Define r : [0, 1] → [0, 1] by r (t) = 1 − t. Then γ ∗ ◦ r = (γ ◦ r )∗ .
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
7/7
Midpoint parameterization
Given a path γ, let m ∈ (0, 1) be such that len(γ[0,m] ) = len(γ[m,1] ).
The midpoint parameterization of γ is the path γ ∗ ∈ [γ] such that
γ ∗ [ 1 ,0] is the parameterization by len of γ[m,0] ; and
2
γ ∗ [ 1 ,1] is the parameterization by len of γ[m,1] .
2
Define r : [0, 1] → [0, 1] by r (t) = 1 − t. Then γ ∗ ◦ r = (γ ◦ r )∗ .
Given two arcs A1 , A2 ⊂ Rn and a bijection f between their endpoint sets,
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
7/7
Midpoint parameterization
Given a path γ, let m ∈ (0, 1) be such that len(γ[0,m] ) = len(γ[m,1] ).
The midpoint parameterization of γ is the path γ ∗ ∈ [γ] such that
γ ∗ [ 1 ,0] is the parameterization by len of γ[m,0] ; and
2
γ ∗ [ 1 ,1] is the parameterization by len of γ[m,1] .
2
Define r : [0, 1] → [0, 1] by r (t) = 1 − t. Then γ ∗ ◦ r = (γ ◦ r )∗ .
Given two arcs A1 , A2 ⊂ Rn and a bijection f between their endpoint sets,
parameterize by paths γ1 , γ2 so that f (γ1 (0)) = γ2 (0), f (γ1 (1)) = γ2 (1).
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
7/7
Midpoint parameterization
Given a path γ, let m ∈ (0, 1) be such that len(γ[0,m] ) = len(γ[m,1] ).
The midpoint parameterization of γ is the path γ ∗ ∈ [γ] such that
γ ∗ [ 1 ,0] is the parameterization by len of γ[m,0] ; and
2
γ ∗ [ 1 ,1] is the parameterization by len of γ[m,1] .
2
Define r : [0, 1] → [0, 1] by r (t) = 1 − t. Then γ ∗ ◦ r = (γ ◦ r )∗ .
Given two arcs A1 , A2 ⊂ Rn and a bijection f between their endpoint sets,
parameterize by paths γ1 , γ2 so that f (γ1 (0)) = γ2 (0), f (γ1 (1)) = γ2 (1).
Then F = γ2∗ ◦ (γ1∗ )−1 is a canonical homeomorphism A1 →A2 extending f .
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
7/7
Midpoint parameterization
Given a path γ, let m ∈ (0, 1) be such that len(γ[0,m] ) = len(γ[m,1] ).
The midpoint parameterization of γ is the path γ ∗ ∈ [γ] such that
γ ∗ [ 1 ,0] is the parameterization by len of γ[m,0] ; and
2
γ ∗ [ 1 ,1] is the parameterization by len of γ[m,1] .
2
Define r : [0, 1] → [0, 1] by r (t) = 1 − t. Then γ ∗ ◦ r = (γ ◦ r )∗ .
Given two arcs A1 , A2 ⊂ Rn and a bijection f between their endpoint sets,
parameterize by paths γ1 , γ2 so that f (γ1 (0)) = γ2 (0), f (γ1 (1)) = γ2 (1).
Then F = γ2∗ ◦ (γ1∗ )−1 is a canonical homeomorphism A1 →A2 extending f .
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
7/7
Midpoint parameterization
Given a path γ, let m ∈ (0, 1) be such that len(γ[0,m] ) = len(γ[m,1] ).
The midpoint parameterization of γ is the path γ ∗ ∈ [γ] such that
γ ∗ [ 1 ,0] is the parameterization by len of γ[m,0] ; and
2
γ ∗ [ 1 ,1] is the parameterization by len of γ[m,1] .
2
Define r : [0, 1] → [0, 1] by r (t) = 1 − t. Then γ ∗ ◦ r = (γ ◦ r )∗ .
Given two arcs A1 , A2 ⊂ Rn and a bijection f between their endpoint sets,
parameterize by paths γ1 , γ2 so that f (γ1 (0)) = γ2 (0), f (γ1 (1)) = γ2 (1).
Then F = γ2∗ ◦ (γ1∗ )−1 is a canonical homeomorphism A1 →A2 extending f .
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
7/7
Midpoint parameterization
Given a path γ, let m ∈ (0, 1) be such that len(γ[0,m] ) = len(γ[m,1] ).
The midpoint parameterization of γ is the path γ ∗ ∈ [γ] such that
γ ∗ [ 1 ,0] is the parameterization by len of γ[m,0] ; and
2
γ ∗ [ 1 ,1] is the parameterization by len of γ[m,1] .
2
Define r : [0, 1] → [0, 1] by r (t) = 1 − t. Then γ ∗ ◦ r = (γ ◦ r )∗ .
Given two arcs A1 , A2 ⊂ Rn and a bijection f between their endpoint sets,
parameterize by paths γ1 , γ2 so that f (γ1 (0)) = γ2 (0), f (γ1 (1)) = γ2 (1).
Then F = γ2∗ ◦ (γ1∗ )−1 is a canonical homeomorphism A1 →A2 extending f .
L. C. Hoehn ([email protected]) (NU)
Paths in Euclidean space
Summer Topology Con. 2014
7/7