Weakly separated self-affine carpets

Abstract

In this paper, we study the Hausdorff and the box-counting dimensions of diagonally aligned self-affine carpets whose projections to the $x$- and $y$-axes satisfy the weak separation condition. In particular, we show that the Hausdorff dimension equals the limit of the Bara\'nski formula, and that the box-counting dimension is the limit of the Feng-Wang formula taken over the $n$-fold compositions of the IFS. We also prove several equivalent formulas for the box-counting dimension, and derive the dimension values for two examples.