Dimension drop of connected part Of slicing self-affine Sponges

TL;DR

This paper investigates the dimension properties of the connected parts of slicing self-affine sponges, showing they either coincide with the entire set or are contained in a smaller attractor, with implications for fractal geometry.

Contribution

It generalizes previous results on fractal cubes to slicing self-affine sponges and demonstrates the dimension drop phenomenon for their connected parts.

Findings

Connected part either equals the set or is contained in a smaller attractor.

Dimension of connected part is strictly less than the whole for Barański carpets.

Removing the slicing property invalidates the main dimension result.

Abstract

The connected part of a metric space $E$ is defined to be the union of non-trivial connected components of $E$. We proved that for a class of self-affine sets called slicing self-affine sponges, the connected part of $E$ either coincides with $E$, or is essentially contained in the attractor of a proper sub-IFS of an iteration of the original IFS.This generalize an early result of Huang and Rao [L. Y. Huang, H. Rao. \emph{A dimension drop phenomenon of fractal cubes}, J. Math. Anal. Appl. \textbf{497} (2021), no. 2] on a class of self-similar sets called fractal cubes. Moreover, we show that the result is no longer valid if the slicing property is removed. Consequently, for a Bara\'nski carpet $E$, the Hausdorff dimension and the box dimension of the connected part of $E$ are strictly less than the Hausdorff dimension and the box dimension of $E$, respectively. For slicing self-affine sponges in $\mathbb R^d$ with $d\geq 3$, whether the attractor of a sub-IFS has strictly smaller dimensions is an open problem.