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

Let A be an n × n real expanding matrix and D be a finite subset of Rn with 0∈D. The family of maps {fd(x)=A−1(x+d)}d∈D is called a self-affine iterated function system (self-affine IFS). The self-affine set K=K(A,D) is the unique compact set determined by (A,D) satisfying the set-valued equation K=⋃d∈Dfd(K). The number s=nln(#D)/ln(q) 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 Hws. In this paper, we show that, if s is chosen to be the pseudo similarity dimension of K, then the condition Hws(K)>0 holds if and only if the IFS {fd}d∈D 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 Hs for a suitable s. Furthermore, we relate the exact value of pseudo Hausdorff measure Hws(K) to a notion of upper s-density with respect to the pseudo norm w(x) associated with A for the measure μ=limM→∞∑d0,...,dM−1∈Dδd0+Ad1+⋯+AM−1dM−1 in the case that #D⩽|detA|.