Self-similar Sets
Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set E is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(E) = E, although the exact definition is given below.
Theorem. Suppose
are contractive mappings on Rn with contraction constant rj < 1. Then there is a unique non-empty compact set A such that
The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of Rn with the Hausdorff distance.
Read more about this topic: Hausdorff Dimension
Famous quotes containing the word sets:
“There be some sports are painful, and their labor
Delight in them sets off. Some kinds of baseness
Are nobly undergone, and most poor matters
Point to rich ends.”
—William Shakespeare (15641616)