Weakly separated self-affine carpets

TL;DR

This paper investigates the Hausdorff and box-counting dimensions of weakly separated self-affine carpets, establishing formulas and limits that generalize previous results for these fractal sets.

Contribution

It provides new formulas and limits for the Hausdorff and box-counting dimensions of weakly separated self-affine carpets, extending existing dimension theory.

Findings

Hausdorff dimension equals the limit of the Baránski formula.

Box-counting dimension is the limit of the Feng-Wang formula over iterated IFS.

Derived explicit dimension values for specific examples.

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.