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
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