5. What is a spacefilling curve?
Definition
A spacefilling curve is a curve fills a region entirely.
Goal: Understand driving functions when transformed into
spacefilling curves.
11. Lip(1
2) functions
Definition
λ is in Lip(1
2) means that there is a c > 0 such that
|λ(t) − λ(s)| ≤ c |t − s|
for all s and t in the domain of λ.
The smallest value of c is called the Lip(1
2) norm of λ.
16. Properties of Lip(1
2) functions
Assume the Lip (1
2) norm of λ(t) = k
1. The Lip (1
2) norm of −λ(t) = k
2. The Lip (1
2) norm of λ(−t) = k
3. The Lip (1
2) norm of λ(t + c) = k
4. The Lip (1
2) norm of λ(t) + c = k
5. The Lip (1
2) norm of cλ(t) = |c|k
6. The Lip (1
2) norm of λ(ct) = k |c|
17. Proof of 1
Lip(1
2) norm is the smallest c such that
|λ(t) − λ(s)| ≤ c |t − s|.
Can be rewritten as:
c = sup {
λ(t) − λ(s)
|t − s|
}
By the definition Lip (1
2) norm, we have
|| − λ||1
2
= sup{
| − λ(t) − (−λ(s))|
|t − s|
}.
Then,
|| − λ||1
2
= sup{
|λ(t) − λ(s)|
|t − s|
}.
Thus,
|| − λ||1
2
= k.
18. Significance of the Lip (1
2) norm
The Lip (1
2) norm gives information about the geometry of a curve.
Theorem (Lind)
If the Lip (1
2) norm is less than 4, we have a simple curve.
Theorem (Lind)
If a curve is spacefilling, then the Lip (1
2) norm is greater than or
equal to 4.0001.
Note: 4.0001 is the wrong number. Our goal is to find out what
this number should be.