The Box-Counting Dimension of Random Box-Like Self-Affine Sets

摘要

In this paper, we study two random analogues of the box-like self-affine attractors introduced by Fraser, which themselves were generalisations of Sierpinski carpets. We determine the almost-sure box-counting dimension for the homogeneous random case (1-variable random), and give a sufficient condition for the almost-sure box-counting dimension to be the expectation of the box-counting dimensions of the corresponding deterministic attractors. Furthermore, we find the almost-sure box-counting dimension of the random recursive model (infinity-variable), which includes affine fractal percolation.