Advantages of the Hausdorff centered measure from the computability point of view

Abstract

In this paper we study Hausdorff centered measures, a useful tool in fractal geometry. The definition of Hausdorff centered measure is based on efficient coverings centered at the given set, playing a dual role to packing measures. We show that, at least in the self-similar setting, it has some advantages, from a computational point of view, over other measures based on coverings, such as the Hausdorff measure or the Hausdorff spherical measure. We also extend our results to general Hausdorff centered measures with gauge functions for which the measures scale.