Self-affine sponges with random contractions

TL;DR

This paper determines the almost sure Hausdorff dimension of random self-affine sponges in high-dimensional space, considering randomness in the defining matrices without requiring separation conditions.

Contribution

It provides a new dimension formula for random self-affine sponges with shared fixed points, removing the need for separation assumptions.

Findings

Derived the Hausdorff dimension formula for random self-affine sponges.

Established results without separation conditions.

Applied to high-dimensional fractal constructions.

Abstract

We compute the almost sure Hausdorff dimension of random self-affine sponges in $\mathbb{R}^d$ without imposing any separation conditions. In this context, randomness arises from the matrices in the defining semigroup, which are random yet the corresponding affine maps share a fixed point.