Lipschitz continuity of the Hausdorff dimension of self-affine sponges at Sierpinski sponges

TL;DR

This paper proves that the Hausdorff dimension of self-affine sponges in three dimensions varies in a Lipschitz continuous manner at Sierpinski sponges, extending known results from two-dimensional carpets.

Contribution

It establishes Lipschitz continuity of the Hausdorff dimension for self-affine sponges at Sierpinski sponges, a novel extension in higher dimensions.

Findings

Hausdorff dimension is Lipschitz continuous at Sierpinski sponges

Extension of dimension continuity results from carpets to sponges

Provides a mathematical foundation for dimension stability in self-affine fractals

Abstract

The Hausdorff dimension of general Sierpinski carpets, [4] and [20], and the generalization on Lalley-Gatzouras carpets, [10], are today well known results, the formulas being obtain via the variational principle for the dimension. We call the multidimensional versions of these carpets Sierpinski sponges and self-affine sponges, respectively,. In this paper we show that the Hausdorff dimension of self-affine sponges, defined in R3, is a Lipschitz continuous function at Sierpinski sponges.