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Abstract: In this paper, we concentrate on the properties of the multifractal centered Hausdorff measure and the multifractal packing measure on a class of cookie-cutter-like (CCL) sets. We conclude that these two multifractal measures are equivalent on the CCL sets satisfying the strong separation condition. Then we obtain some relevant conclusions about the image measure of a -invariant ergodic Borel probability measure. [Copyright &y& Elsevier]

Abstract: We discuss the fractal dimension and measure for the subset B P (ω) of Moran set E(ω) in R d satisfying the strong separation condition. Firstly, we give the Hausdorff dimension of subset B P (ω) in compatible case and incompatible case. Then we attain that there exists a subset B of the set B P (ω) such that B has full μ P -measure but zero Hausdorff measure in incompatible case. Finally, if the gap condition holds, we see that B P (ω) and E(ω) have the same Hausdorff measure and packing measure, and both of them are α-sets in compatible case. [Copyright &y& Elsevier]

Abstract: For 3-part Cantor set, we first discuss the relationship between iterated function systems and the union of the set with its two translations. Then we obtain the necessary and sufficient condition that the union is a self-similar set with the open set condition. [Copyright &y& Elsevier]

Abstract: Motivated by Mandelbrot’s idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, Nekka and Li studied the properties of these translation sets and showed how they can be used for the classification purpose. In this paper, we pursue this study on a class of Moran sets with their rational translates. We also get the fractal structure of intersection I(x, y) of a class of Moran sets with their rational translates, and the formula of the box-counting dimension. We find that the Hausdorff measures of these sets form a discrete spectrum whose non-zero values come only from shifting vector with the expansion in fraction of (x, y). Concretely, when (x, y) has a finite expansion in fraction, a very brief calculation formula of the measure is given. [Copyright &y& Elsevier]

Abstract: Motivated by Mandelbrot’s idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, Nekka and Li studied the properties of these translation sets and showed how they can be used for a classification purpose. In this paper, we pursue this study on the Sierpinski carpet with its rational translate. We also get the fractal structure of intersection I(x, y) of the Sierpinski carpet with its translate. We find that the packing measure of these sets forms a discrete spectrum whose non-zero values come only from shifting numbers with a finite triadic expansion. Concretely, when x and y have a finite triadic expansion, a very brief calculation formula of the measure is given. [Copyright &y& Elsevier]

Abstract: We investigate a class of Moran sets in general metric space. For the Moran sets satisfying the strong separation condition we mainly discuss the equivalence of the centered Hausdorff measure and the Packing measure. An actual example of fractal sets satisfying the above property is given. [Copyright &y& Elsevier]

Abstract: Let K(λ 1, λ 2), the symmetry Cantor sets, be the attractor of an iterated function system {f 1, f 2, f 3} on the line, where f 1(x)= λ 1 x, , x ∈[0,1]. In this paper, we proved that if , where λ ≡max{λ 1, λ 2}, then the exact Hausdorff centered measure C s of K(λ 1, λ 2) equals 1, where s is the Hausdorff dimension of K(λ 1, λ 2). [Copyright &y& Elsevier]