Dimensions of a class of nonautonomous carpets and measures on $\R^{2}$

Abstract

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_{k=1}^{\infty}\bigcup_{(w_1w_2\ldots w_k)\in \prod_{i=1}^k\mathcal{D}_i}\Phi_{w_1}\circ \Phi_{w_2}\circ \cdots\circ \Phi_{w_k} is called a nonautonomous carpet .In the paper, we provide the lower, packing, box-counting and Assouad dimensions of the nonautonomous carpets E . We also explore the dimension properties of nonautonomous measures \mu supported on E , and we provide Hausdorff, packing and entropy dimension formulas of \mu .