, Volume 74, Issue 3, pp 357-392

Statistically self-similar fractals

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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 setBK(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, ..., SN-1), eachS i has an inverse which satisfies a Lipschitz condition then there is an α≧0 such that, forP μ-a.e.KK(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, ..., SN-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.KK(X) equals O.

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

Supported by a Heisenberg grant of the Deutsche Forschungsgemeinschaft