Packing dimension and measure of homogeneous Cantor sets

Abstract

For a class of homogeneous Cantor sets, we find an explicit formula for their packing dimensions. We then turn our attention to the value of packing measures. The exact value of packing measure for homogeneous Cantor sets has not yet been calculated even though that of Hausdorff measures was evaluated by Qu, Rao and Su in (2001). We give a reasonable lower bound for the packing measures of homogeneous Cantor sets. Our results indicate that duality does not hold between Hausdorff and packing measures.

Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure

“…Introduced by Tricot [33], the concept of Packing measure and dimension for fractals was studied by several authors (see e.g. [2,[6][7][8] and references therein). Let ε > 0 and I ⊂ N. An ε-packing of F is a collection of disjoint balls (B i ) i∈I with diameter at most ε and midpoints of B i placed in F .…”

“…First let us outline with the following two theorems a part of these known results, which are needed in this paper. [2,38].) Assume that condition (BD) is satisfied.…”

“…Beside of the known linkages between quantization, Hausdorff respectively Packing dimension (cf. [2,10,15,20,29,38]), we will show that the Euler exponent equals the quantization dimension under certain restrictions (cf. Theorems 2.5, 2.7).…”

“…As a third aspect, the Hausdorff and Packing measure concentrated on fractals respectively the dimensions based on these measures were also studied intensively (see e.g. [2,7,17,19,31]). The aim of this paper is to present new coherences between these three aspects in the theory of fractals for the special case of one-dimensional homogeneous Cantor sets.…”

The Correlation Dimension of Generalized Cantor-Like Sets

“…(cf, [1,2,4,5,13,14,19]). The author in [1] found out lower and upper bounds for the Hausdorff dimension of a deranged Cantor set and the authors in [2] and [19] calculated the exact packing dimension and Hausdorff dimension, respectively. In [4], they found the Hausdorff dimension of a Cantor set considering infinitely many contractive ratios.…”