The box dimension of random box-like self-affine sets

TL;DR

This paper investigates the box-counting dimension of two types of random self-affine fractals, extending previous deterministic models, and provides conditions under which their dimensions are almost surely determined.

Contribution

It introduces the analysis of the box dimension for homogeneous and recursive random self-affine sets, including affine fractal percolation, with new almost sure dimension results.

Findings

Almost sure box dimension for homogeneous random case determined.

Sufficient condition for the dimension to equal the expected deterministic dimensions.

Almost sure box dimension established for the recursive random model including affine fractal percolation.

Abstract

In this paper we study two random analogues of the box-like self-affine attractors introduced by Fraser, itself an extension of Sierpi\'nski 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 dimension to be the expectation of the box dimensions of the deterministic attractors. Furthermore we find the almost sure box-counting dimension of the random recursive model ($\infty$-variable), which includes affine fractal percolation.