A MATRIX METHOD FOR APPROXIMATING FRACTAL MEASURES

Abstract

Let X be a bounded subset of Rn and let A be the Lebesgue measure on X. Let {X:τ1,…, τN} be an iterated function system (IFS) with attractor S. We associate probabilities p1,…, pN with τ1,…, τN, respectively. Let M(X) be the space of Borel probability measures on X, and let M: M(X)→M(X) be the Markov operator associated with the IFS and its probabilities given by: [Formula: see text] where A is a measurable subset of X. Then there exists a unique µ∈M (A) such that Mµ=µ; µ is referred to as the measure invariant under the iterated function system with the associated probabilities. The support of μ is the attractor S. We prove the existence of a sequence of step functions {fi}, which are the eigenvectors of matrices {Mi}, such that the measures {fidλ} converge weakly to µ. An algorithm is presented for the construction of Mi and an example is given.