Dimensions of a class of self-affine Moran sets and measures in ${\mathbb{R}}^2$

Abstract

For each integer $k>0$, let $n_k$ and $m_k$ be integers such that $n_k\ge 2,m_k\ge 2$, and let ${\mathcal{D}}_k$ be a subset of $\{0,\dots ,n_k-1\}\times \{0,\dots ,m_k-1\}$. For each $w=(i,j)\in {\mathcal{D}}_k$, we define an affine transformation on~${\mathbb{R}}^2$ by $${\Phi }_w(x)=T_k(x+w),w\in {\mathcal{D}}_k,$$ where $T_k=\mathrm{diag}(n_k^{-1},m_k^{-1})$. The non-empty compact set $$E={\bigcap }_{k=1}^{\infty }{\bigcup }_{(w_1{w}_2\dots w_k)\in {\prod }_{i=1}^k{\mathcal{D}}_i}{\Phi }_{w_1}\circ {\Phi }_{w_2}\circ \dots \circ {\Phi }_{w_k}$$ is called a \textit{self-affine Moran set}. In the paper, we provide the lower, packing, box-counting and Assouad dimensions of the self-affine Moran set $E$. We also explore the dimension properties of self-affine Moran measure $\mu$ supported on $E$, and we provide Hausdorff, packing and entropy dimension formulas of $\mu$.