Probability Theory and Related Fields

, Volume 74, Issue 3, pp 357–392

# Statistically self-similar fractals

• Siegfried Graf
Article

DOI: 10.1007/BF00699096

Graf, S. Probab. Th. Rel. Fields (1987) 74: 357. doi:10.1007/BF00699096

## 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 measurePμ 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. (S0, ..., SN-1), eachSi has an inverse which satisfies a Lipschitz condition then there is an α≧0 such that, forPμ-a.e.KK(X), the Hausdorff dimensionH-dim(K) is equal to α. IfX⊂ℝd is compact and has non-empty interior and if μ-a.e. (S0, ..., 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(Si) denotes the (smallest) Lipschitz constant forSi. Under fairly general assumptions the α-dimensional Hausdorff measure ofPμ-a.e.KK(X) equals O.

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