| Now we shall prove that if all the Ti are euclidean contractions, then
T is a Hausdorff contraction. |
| First we need a Lemma. |
| Lemma 1 For all compact sets A, B, C, and D in the plane, |
| h(A ∪ B, C ∪ D) ≤ max{h(A,C),h(B,D)} |
| Here is the proof of the lemma. |
| Now we are ready to prove the the proposition. |
| Prop. 2 For all compact sets A and B in the plane, |
| h(T (A),T (B)) ≤ r⋅h(A,B) |
| where r = max{r1,...,rN}, and ri is the
euclidean contraction factor of Ti. |
| Proof |
| h(T (A),T (B)) |
| = h(T1(A) ∪ T2(A) ∪ ... ∪ TN(A),
T1(B) ∪ T2(B) ∪ ... ∪ TN(B)) |
| = h(T1(A) U (T2(A) ∪ ... ∪ TN(A)),
T1(B) ∪ (T2(B) ∪ ... ∪ TN(B))) |
| separating T1(A) from Ti(A),
and T1(B) from Ti(B) |
| ≤ max{h(T1(A),T1(B)),
h(T2(A) ∪ ... ∪ TN(A),
T2(B) ∪ ... ∪ TN(B))} |
| by the Lemma |
| ≤ max{h(T1(A),T1(B)),
h(T2(A),T2(B)),
..., h(TN(A),TN(B))} |
| keep applying the Lemma |
| ≤ max{r1⋅h(A,B),
r2⋅h(A,B), ...,
rN⋅h(A,B)} |
| by Prop. 1 |
| = max{r1, r2, ..., rN}⋅h(A,B) |
| = r⋅h(A,B) |
|