Densities of self-similar measures on the line

Abstract

We describe algorithms to compute self-similar measures associated to iterated function systems (i.f.s.) on an interval, and more general self-replicating measures that include Hausdorff measure on the attractor of a nonlinear i.f.s. We discuss a variety of error measurements for these algorithms. We then use the algorithms to study density properties of these measures experimentally. By density we mean the behavior of the ratio $\mu(B_r(x))/(2r)^\alpha$ as $r \rightarrow 0$, were $\alpha$ is an appropriate dimension. It is well-known that a limit usually does not exist. We have found an intriguing structure associated to these ratios that we call density diagrams. We also use density computations to approximate the exact Hausdorff measure of the attractor of an i.f.s.