Calculating Hausdorff dimension in higher dimensional spaces

TL;DR

This paper establishes a formula linking the Hausdorff dimension of subsets in higher-dimensional spaces to that of their preimages in one dimension, enabling easier calculation of fractal dimensions in complex spaces.

Contribution

It introduces a new identity for Hausdorff dimension in higher dimensions using fractal structures, generalizing previous theorems and facilitating practical dimension calculations.

Findings

Proves $ ext{dim}_H(F)=d imes ext{dim}_H(eta^{-1}(F))$ for subsets of $ extbf{R}^d$.

Generalizes Skubalska-Rafaj extbackslash{}l{}owicz and García-Mora-Redtwitz theorems.

Enables effective fractal dimension calculation in applications.

Abstract

In this paper, we prove the identity $\dim_{\textrm H}(F)=d\cdot \dim_{\textrm H}(\alpha^{-1}(F))$, where $\dim_{\textrm H}$ denotes Hausdorff dimension, $F\subseteq \mathbb{R}^d$, and $\alpha:[0,1]\to [0,1]^d$ is a function whose constructive definition is addressed from the viewpoint of the powerful concept of a fractal structure. Such a result stands particularly from some other results stated in a more general setting. Thus, Hausdorff dimension of higher dimensional subsets can be calculated from Hausdorff dimension of $1-$dimensional subsets of $[0,1]$. As a consequence, Hausdorff dimension becomes available to deal with the effective calculation of the fractal dimension in applications by applying a procedure contributed by the authors in previous works. It is also worth pointing out that our results generalize both Skubalska-Rafaj\l{}owicz and Garc\'{\i}a-Mora-Redtwitz theorems.