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

Abstract

Let A be an n × n real expanding matrix and D be a finite subset of R n with 0 ∈ D. The family of maps {f d (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∈D f d (K). The number s = n ln(#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 H s w . In this paper, we show that, if s is chosen to be the pseudo similarity dimension of K, then the condition H s w (K) > 0 holds if and only if the IFS {f d } 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 H s for a suitable s. Furthermore, we relate the exact value of pseudo Hausdorff measure H s w (K) to a notion of upper s-density with respect to the pseudo norm w(x) associated with A for the measure µ = lim M→∞ d0,...,dM−1∈D δ d0+Ad1+···+A M −1 dM−1 in the case that #D ≤ |det A|.