The Hausdorff dimension of self-affine Sierpinski sponges

TL;DR

This paper calculates the Hausdorff dimension of certain 3D self-affine fractals and demonstrates that a variational principle applies to these structures, advancing understanding of their geometric complexity.

Contribution

It provides an explicit computation of the Hausdorff dimension for a class of self-affine Sierpinski sponges and confirms the validity of the variational principle for these sets.

Findings

Hausdorff dimension computed for specific self-affine sets

Validation of the variational principle for this class

Enhanced understanding of fractal geometric properties

Abstract

We compute the Hausdorff dimension of limit sets generated by 3-dimensional self-affine mappings with diagonal matrices of the form A_{ijk}=Diag(a_{ijk}, b_{ij}, c_{i}), where 0<a_{ijk}\le b_{ij}\le c_i<1. By doing so we show that the variational principle for the dimension holds for this class.