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28 March 2024
 
  » arxiv » 1903.2394

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Open set condition and pseudo Hausdorff measure of self-affine IFSs
Xiaoye Fu ; Jean-Pierre Gabardo ; Hua Qiu ;
Date 28 Feb 2019
AbstractLet $A$ be an $n imes n$ real expanding matrix and $mathcal{D}$ be a finite subset of $mathbb{R}^n$ with $0inmathcal{D}$. The family of maps ${f_d(x)=A^{-1}(x+d)}_{dinmathcal{D}}$ 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 $K=displaystyleigcup_{dinmathcal{D}}f_d(K)$. 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 $sge 0$ 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 ${f_d}_{dinmathcal{D}}$ 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 $mu=limlimits_{M oinfty}sumlimits_{d_0,dotsc,d_{M-1}inmathcal{D}}delta_{d_0 + Ad_1 + dotsb + A^{M-1}d_{M-1}}$ in the case that $#mathcal{D}lelvertdet A vert$.
Source arXiv, 1903.2394
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