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

Abstract

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 $K=\bigcup _{d\in \mathcal{D}}{f}_d(K)$. The number $s=n\ln (\#\mathcal{D})/\ln (q)$ 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 $s\ge 0$ a natural pseudo Hausdorff measure denoted by ${\mathcal{H}}_w^s.$ 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 $\mu =\underset{M\to \infty }{\lim }\sum _{d_0,\dots ,d_{M-1}\in \mathcal{D}}{\delta }_{d_0+A{d}_1+\cdots +A^{M-1}{d}_{M-1}}$ in the case that $\#\mathcal{D}\le |\det A|$.

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