• Title/Summary/Keyword: deranged Cantor set

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HAUSDORFF DIMENSION OF DERANGED CANTOR SET WITHOUT SOME BOUNDEDNESS CONDITION

  • Baek, In-Soo
    • Communications of the Korean Mathematical Society
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    • v.19 no.1
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    • pp.113-117
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    • 2004
  • A deranged Cantor set (without the uniform bounded-ness condition away from zero of contraction ratios) whose weak local dimensions for all points coincide has its Hausdorff dimension of the same value of weak local dimension. We will show it using an energy theory instead of Frostman's density lemma which was used for the case of the deranged Cantor set with the uniform boundedness condition of contraction ratios. In the end, we will give an example of such a deranged Cantor set.

DIMENSIONS OF A DERANGED CANTOR SET WITH SPECIFIC CONTRACTION RATIOS

  • Baek, In-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.2
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    • pp.269-274
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    • 2004
  • We investigate a deranged Cantor set (a generalized Cantor set) using the similar method to find the dimensions of cookie-cutter repeller. That is, we will use a Gibbs measure which is a weak limit of a subsequence of discrete Borel measures to find the dimensions. The deranged Cantor set that will be considered is a generalized form of a perturbed Cantor set (a variation of the symmetric Cantor set) and a cookie-cutter repeller.

CANTOR DIMENSION AND ITS APPLICATION

  • Baek, In-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.1
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    • pp.13-18
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    • 2004
  • We defined Cantor dimensions of a perturbed Cantor set, and investigated a relation between these dimensions and Hausdorff and packing dimensions of a perturbed Cantor set. In this paper, we introduce another expressions of the Cantor dimensions. Using these, we study some informations which can be derived from power equations induced from contraction ratios of a perturbed Cantor set to give its Hausdorff or packing dimension. This application to a deranged Cantor set gives us an estimation of its Hausdorff and packing dimensions, which is a generalization of the Cantor dimension theorem.

RELATION BETWEEN FRACTAL MEASURES AND CANTOR MEASURES

  • Baek, In-Soo
    • Communications of the Korean Mathematical Society
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    • v.22 no.2
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    • pp.241-246
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    • 2007
  • We investigate the relation between Hausdorff(packing) measure and lower(packing) Cantor measure on a deranged Cantor set. If the infimum of some distortion of contraction ratios is positive, then Hausdorff(packing) measure and lower(packing) Cantor measure of a deranged Cantor set are equivalent except for some singular behavior for packing measure case. It is a generalization of already known result on the perturbed Cantor set.

ON A SELF-SIMILAR MEASURE ON A SELF-SIMILAR CANTOR SET

  • Baek, In-Soo
    • Journal of the Chungcheong Mathematical Society
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    • v.16 no.2
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    • pp.1-10
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    • 2003
  • We compare a self-similar measure on a self-similar Cantor set with a quasi-self-similar measure on a deranged Cantor set. Further we study some properties of a self-similar measure on a self-similar Cantor set.

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MULTIFRACTAL BY SELF-SIMILAR MEASURES

  • Baek, In-Soo
    • Journal of applied mathematics & informatics
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    • v.23 no.1_2
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    • pp.497-503
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    • 2007
  • We consider a non-empty subset having same local dimension of a self-similar measure on a most generalized Cantor set. We study trans-formed lower(upper) local dimensions of an element of the subset which are local dimensions of all the self-similar measures on the most generalized Cantor set. They give better information of Hausdorff(packing) dimension of the afore-mentioned subset than those only from local dimension of a given self-similar measure.