Random Survivor Sets with Moving Holes in Bedford-McMullen Carpets

Abstract

Let $2\le m < n$ be integers, and let $K(\mathcal{A})$ be the Bedford--McMullen carpet generated by $\{(x,y)\mapsto((x+i)/m,(y+j)/n):(i,j)\in\mathcal{A}\}$, where $\mathcal{A}$ is a subset of $\{0,1,\ldots,m-1\}\times\{0,1,\ldots,n-1\}$. Write $\mathcal{B}$ for the set of occupied columns, and assume that $\#\mathcal{B}\ge2$ and each column contains at least two symbols of $\mathcal{A}$. At each vertex of the column tree $\mathcal{B}^*$, we independently choose one forbidden symbol from $\mathcal{A}$ according to a prescribed distribution. Let $\mathbb{P}$ denote the resulting product measure on $\mathcal{A}^{\mathcal{B}^*}$. This gives a random survivor set $K_ω(\mathcal{A})$ with $ω\in\mathcal{A}^{\mathcal{B}^*}$. We show that, for $\mathbb{P}$-a.e. $ω$, \begin{equation*} \dim_H K_ω(\mathcal{A})=\frac{1}{\log m}\log\left(\int \sum_{b\in\mathcal{B}}\ell_b(ω)^{\frac{\log m}{\log n}} d\mathbb{P}(ω)\right), \end{equation*} where $\ell_b(ω)$ is the number of surviving symbols in column $b$ at the root.