Iterated function systems and the global construction of fractals

Abstract

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.