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

Let $A$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of ${\mathbb{R}}^n$ with $0\in \mathcal{D}$. The family of maps $\{f_d(x)=A^{-1}(x+d){\}}_{d\in \mathcal{D}}$ is called a self-affine iterated function system (self-affine IFS). The self-affine set $K=K(A,\mathcal{D})$ is the unique compact set determined by $(A,\mathcal{D})$ satisfying the set-valued equation $K={\displaystyle \bigcup _{d\in \mathcal{D}}{f}_d(K)}$. The number $s=n\ln (\mathrm{\#}\mathcal{D})\mathrm{/}\ln (q)$ with $q=\mathrm{\mid }\det (A)\mathrm{\mid }$, is the so-called pseudo similarity dimension of $K$. As shown by He and Lau, one can associate with $A$ and any number $s\ge 0$ a natural pseudo Hausdorff measure denoted by ${\mathcal{H}}_w^s\mathrm{.}$ In this paper, we show that, if $s$ is chosen to be the pseudo similarity dimension of $K$, then the condition ${\mathcal{H}}_w^s(K)>0$ holds if and only if the IFS $\{f_d{\}}_{d\in \mathcal{D}}$ 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 ${\mathcal{H}}^s$ for a suitable $s$. Furthermore, we relate the exact value of pseudo Hausdorff measure ${\mathcal{H}}_w^s(K)$ to a notion of upper $s$-density with respect to the pseudo norm $w(x)$ associated with $A$ for the measure $\mu=\lim\limits_{M\to\infty}\sum\limits_{d_0,\dotsc,d_{M-1}\in$\mathcal{D}$}\delta_{d_0

• Ad_1 + \dotsb + A^{M-1}d_{M-1}}$inthecasethat$#$\mathcal{D}$\le\lvert\det A\rvert$.