Computability of the Hausdorff and packing measures on self-similar sets and the self-similar tiling principle

Abstract

We state a self-similar tiling principle which shows that any open subset of a self-similar set with open set condition may be tiled without loss of measure by copies under similitudes of any closed subset with positive measure. We use this method to get the optimal coverings and packings which give the exact value of the Hausdorff-type and packing measures. In particular, we show that the exact value of these measures coincides with the supremum or with the infimum of the inverse of the density of the natural probability measure on suitable classes of sets. This gives criteria for the numerical analysis of the measures, and allows us to compare their complexity in terms of computability.

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

“…To do so, we consider first the case when F ⊂ O and then we extend (11) to general closed sets. We start by repeating the construction of Lemma 4 in [23] to get a δ-tiling J ∈ U : F from the δ-tilings I ∈ U : E and…”

“…The point of view is entirely different from Edgar's and instead it follows the lines of [23] where similar expressions are obtained for the Hausdorff and packing measures. We first show that the Hausdorff centered measure C s (E) of a self-similar set E coincides with its premeasure C s 0 (E), making unnecessary the second step in its definition above.…”

“…The packing measure has been extensively studied in the later years. In a self-similar setting, the packing measure may be easier to handle than the Hausdorff measure [23], however, it is not, in a general setting, equivalent to the Hausdorff measure in the sense that a set of null s-dimensional Hausdorff measure may well have an infinite s-dimensional packing measure. Moreover, the behaviour of the packing measure under natural operations such as projections or intersections is very irregular unlike the one of the Hausdorff (see, for example, [6], [7], [8], [9], [12], [13], [16], [19], [20].)…”

“…Surprisingly, as Theorem 3 shows, on compact subsets of self similar sets satisfying the open set condition (OSC), we are able to show that the second step is not needed. The proof of Theorem 3 is based on the self-similar tiling principle proved by the second author in [23].…”

“…Notice that the fact that the centers of the balls belong to E makes easier the search of the ball of minimal density in comparison with the related result for the Hausdorff spherical measure. For the Hausdorff spherical measure the optimal balls could be centered anywhere in R n [23]. The computation of the ordinary Hausdorff measure implies a greater degree of complexity, since it requires the search of a convex set of minimal density.…”