Iterated function systems and the global construction of fractals. (English) Zbl 0588.28002

Barnsley, M. F.; Demko, Stephen

Proc. R. Soc. Lond., Ser. A 399, 243-275 (1985).

Authors’ abstract: Iterated function systems (i.f.ss) are introduced as a unified way of generating a broad class of fractals. These fractals are often attractors for i.f.ss and occur as the supports of probability measures associated with functional equations. The existence of certain ’p-balanced’ measures for i.f.ss is established, and these measures are uniquely characterized for hyperbolic i.f.ss. The Hausdorff-Besicovitch dimension for some attractors of hyperbolic i.f.ss is estimated with the aid of p-balanced measures. What appears to be the broadest framework for the exactly computable moment theory of p-balanced measures - that of linear i.f.ss and of probabilistic mixtures of iterated Riemann surfaces - is presented. This extensively generalizes earlier work on orthogonal polynomials on Julia sets. An example is given of fractal reconstruction with the use of linear i.f.ss and moment theory.

Reviewer: O.Lipovan

MSC:

28A75

Length, area, volume, other geometric measure theory

Keywords:

Iterated function systems; fractals; attractors; Hausdorff-Besicovitch dimension; p-balanced measures; probabilistic mixtures of iterated Riemann surfaces; orthogonal polynomials on Julia sets