Now we relate the Euclidean contraction factor of |
T:R2 → R2 |
to the Hausdorff contraction factor of |
T:K(R2) → K(R2) |
defined by T(A) = {T(x,y): for each (x,y) in A}. |
Prop. 1 If T has Euclidean contraction factor r, then for all compact sets
A and B in the plane |
h(T(A),T(B)) ≤ r⋅h(A,B) |
Proof Let h(A,B) = k. We want to show |
T(A) ⊆ (T(B))r⋅k |
and |
T(B) ⊆ (T(A))r⋅k |
That is, for every point a' of T(A) there is a point
b' of T(B) with d(a',b') ≤ r⋅k. The
analogous result must be shown to establish the other inclusion. |
 |
For every point a' in T(A), there is some point
a in A with T(a) = a'. |
Because A is contained in Bk, there is some point
b in B with d(a, b) ≤ k. |
Let b' = T(b). Certainly, b' is in T(B). |