A MATRIX METHOD FOR APPROXIMATING FRACTAL MEASURES

Let X be a bounded subset of Rⁿ and let A be the Lebesgue measure on X. Let {X:τ₁,…, τ_N} be an iterated function system (IFS) with attractor S. We associate probabilities p₁,…, p_N with τ₁,…, τ_N, 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:

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 {f_i}, which are the eigenvectors of matrices {M_i}, such that the measures {f_idλ} converge weakly to µ. An algorithm is presented for the construction of M_i and an example is given.