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 $\mu$.
Categories: math.CA