Dimension drop of connected part of slicing self-affine sponges

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 generalizes an early result of Huang and Rao (2021) [9] 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 Moreover, we show that the result is no longer valid if the slicing property is removed.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 Barański carpet E possessing trivial points, the Hausdorff dimension and the box dimension of the connected part of E are strictly less than that of E, respectively. For slicing self-affine sponges in Rd with d≥3, whether the attractor of a sub-IFS has strictly smaller dimensions is an open problem.