Dimensions of a class of self-affine Moran sets and measures in \(\R^2\)

For each integer \(k>0\), let \(n_k\) and \(m_k\) be integers such that \(n_k\geq 2, m_k\geq 2\), 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~\(\R^2\) by $$ \Phi_w(x)=T_k(x+w), \qquad w\in\mathcal{D}_k, $$ where \(T_k=\operatorname{diag}(n_k^{-1},m_k^{-1})\). The non-empty compact set $$ E=\bigcap\nolimits_{k=1}^{\infty}\bigcup\nolimits_{(w_1w_2\ldots w_k)\in \prod_{i=1}^k\mathcal{D}_i} \Phi_{w_1}\circ \Phi_{w_2}\circ \ldots\circ \Phi_{w_k} $$ 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 \(\mu\).