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 |
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For every point |
Return to Convergence of determinisitc IFS.