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Some properties for the intersection of Moran sets with their translates

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  • Dai, Meifeng
  • Tian, Lixin

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.

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  • Dai, Meifeng & Tian, Lixin, 2007. "Some properties for the intersection of Moran sets with their translates," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 757-764.
    • Handle: RePEc:eee:chsofr:v:31:y:2007:i:3:p:757-764
    DOI: 10.1016/j.chaos.2006.03.074
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    References listed on IDEAS

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    1. El Naschie, M.S., 2006. "Elementary number theory in superstrings, loop quantum mechanics, twistors and E-infinity high energy physics," Chaos, Solitons & Fractals, Elsevier, vol. 27(2), pages 297-330.
    2. Dai, Meifeng, 2006. "The equivalence of measures on Moran set in general metric space," Chaos, Solitons & Fractals, Elsevier, vol. 29(1), pages 55-64.
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    Cited by:

    1. Dai, Meifeng & Tian, Lixin, 2008. "On the intersection of an m-part uniform Cantor set with its rational translation," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 962-969.

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