A dimension drop phenomenon of fractal cubes

Abstract

Let E be a metric space. We introduce a notion of connectedness index of E, which is the Hausdorff 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 Hausdorff 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 Hausdorff dimension.