Open set condition and pseudo Hausdorff measure of self-affine IFSs

Abstract

Let A be an n × n real expanding matrix and be a finite subset of with . The family of maps is called a self-affine iterated function system (self-affine IFS). The self-affine set is the unique compact set determined by satisfying the set-valued equation . The number with q = |det(A)|, is the so-called pseudo similarity dimension of K. As shown by He and Lau, one can associate with A and any number s ⩾ 0 a natural pseudo Hausdorff measure denoted by . In this paper, we show that, if s is chosen to be the pseudo similarity dimension of K, then the condition holds if and only if the IFS satisfies the open set condition (OSC). This extends the well-known result for the self-similar case that the OSC is equivalent to K having positive Hausdorff measure for a suitable s. Furthermore, we relate the exact value of pseudo Hausdorff measure to a notion of upper s-density with respect to the pseudo norm w(x) associated with A for the measure in the case that .