Fractals in the Large

Abstract

Abstract A reverse iterated function system (r.i.f.s.) is defined to be a set of expansive maps ﹛T 1 ,…, T m ﹜ on a discrete metric space M. An invariant set F is defined to be a set satisfying , and an invariant measure μ is defined to be a solution of for positive weights p j . The structure and basic properties of such invariant sets and measures is described, and some examples are given. A blowup ℱ of a self-similar set F in ℝ n is defined to be the union of an increasing sequence of sets, each similar to F. We give a general construction of blowups, and show that under certain hypotheses a blowup is the sum set of F with an invariant set for a r.i.f.s. Some examples of blowups of familiar fractals are described. If μ is an invariant measure on ℤ + for a linear r.i.f.s., we describe the behavior of its analytic transform, the power series on the unit disc.