Statistically self-similar fractals

Summary

LetX be a complete separable bounded metric space and μ a Borel probability measure on the space Con(X)^N of allN-tuples of contractions ofX with the topology of pointwise convergence. Then there exists a unique μ-self-similar probability measureP _μ on the spaceK(X) of all non-empty compact subsets ofX. Here a measureP onK(X) is called μ-self-similar if, for every Borel setB ⊂K(X),

$$P(B) = \int {P^N } \left( {(K_O ,...,K_{N - 1} )\left| {\bigcup\limits_{i = 0}^{N - 1} {S_i (K_i ) \in B} } \right.} \right)d\mu (S_O ,...,S_{N - 1} ).$$

If, for μ-a.e. (S ₀, ..., S_N-1), eachS _i has an inverse which satisfies a Lipschitz condition then there is an α≧0 such that, forP _μ-a.e.K∈K(X), the Hausdorff dimensionH-dim(K) is equal to α. IfX⊂ℝ _d is compact and has non-empty interior and if μ-a.e. (S ₀, ..., S_N-1) consists of similarities which satisfy a certain disjointness condition w.r.t.X then α is determined by the equation

$$\int {\sum\limits_{i = 0}^{N - 1} {Lip(S_i )^\alpha } } d\mu (S_O ,...,S_{N - 1} ) = 1,$$

where Lip(S _i) denotes the (smallest) Lipschitz constant forS _i. Under fairly general assumptions the α-dimensional Hausdorff measure ofP _μ-a.e.K∈K(X) equals O.

If μ andX are chosen in a rather special way thenP _μ-a.e.K∈K(X) is the graph of a homeomorphism of [0, 1] (or a curve or the graph of a continuous function).