Dimension drop of connected part of slicing self-affine sponges (Q2069762)

From the authors' abstract:: The connected part of a metric space \((X,d)\) is defined to be the union of non-trivial connected components of \(X\). The authors proved that for a class of self-affine sets called slicing self-affine sponges, it is always homeomorphic to its associated Sierpiński self-affine sponge, or is essentially contained in the attractor of a proper sub-IFS of an iteration of the original IFS, which is a generalisation of an early result of \textit{L.-y. Huang} and \textit{H. Rao} [J. Math. Anal. Appl. 497, No. 2, Article ID 124918, 12 p. (2021; Zbl 1461.28004)] on a class of self-similar sets called fractal cubes. Moreover, the authors showed that the result is no longer valid if the slicing property is removed. Consequently, for a Barański carpet possessing trivial points, the authors obtained that the Hausdorff dimension and the box dimension of the connected part of such Barański carpet are strictly less than that of the Barański carpet respectively. Moreover, it has been remained as an open problem whether the attractor of a sub-IFS has strictly smaller dimensions for slicing self-affine sponges in \(R^n\) with \(n \geq 3\).