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.
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