Hausdorff Dimension - Self-similar Sets

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 r_j < 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.