A dimension drop phenomenon of fractal cubes

TL;DR

This paper introduces a new invariant called the connectedness index for metric spaces, particularly fractal cubes, showing it is strictly less than the Hausdorff dimension when a trivial connected component exists, revealing a dimension drop phenomenon.

Contribution

It defines the connectedness index for metric spaces and demonstrates its properties and relation to other dimensions, especially in fractal cubes, as a new Lipschitz invariant.

Findings

Connectedness index is strictly less than Hausdorff dimension for certain fractal cubes.

Connectedness index serves as a new Lipschitz invariant.

Explores the relationship between connectedness index and topological Hausdorff dimension.

Abstract

Let E be a metric space. We introduce a notion of connectedness index of E, which is the Hausdor? dimension of the union of non-trivial connected components of E. We show that the connectedness index of a fractal cube E is strictly less than the Hausdor? dimension of E provided that E possesses a trivial connected component. Hence the connectedness index is a new Lipschitz invariant. Moreover, we investigate the relation between the connectedness index and topological Hausdor? dimension.