• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1737-1751
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• 2015

For every doubling gauge g, we prove that there is a Cantor set of positive finite $H^g$-measure, $P^g$-measure, and $P^g_0$-premeasure. Also, we show that every compact metric space of infinite $P^g_0$-premeasure has a compact countable subset of infinite $P^g_0$-premeasure. In addition, we obtain a class of uniform Cantor sets and prove that, for every set E in this class, there exists a countable set F, with $\bar{F}=E{\cup}F$, and a doubling gauge g such that $E{\cup}F$ has different positive finite $P^g$-measure and $P^g_0$-premeasure.

• The Pure and Applied Mathematics
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• v.14 no.1 s.35
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• pp.1-5
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• 2007

We study the topological magnitude of a special subset from the distribution subsets in a self-similar Cantor set. The special subset whose every element has no accumulation point of a frequency sequence as some number related to the similarity dimension of the self-similar Cantor set is of the first category in the self-similar Cantor set.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.397-403
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• 2004

Cutler showed some duality results about Hausdorff and packing dimensions of a measure on a compact set in Euclidean space if its s-dimensional Hausdorff measure or packing measure is positive. We show that the similar results in a perturbed Cantor set hold according to its quasi s-dimensional Hausdorff measure or packing measure and we find concrete measures in this case while Cutler showed the existence of such measures. Finally under some strong condition, we give a concrete measure whose Hausdorff and packing dimensions are the same as those of the perturbed Cantor set without the condition that it has positive s-dimensional Hausdorff or packing measures.

• Communications of the Korean Mathematical Society
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• v.16 no.1
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• pp.95-102
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• 2001

We consider a Cantor set with overlaps Λ in R$^1$. We calculate its correlation dimension with respect to the push-down measure on Λ comparing with its similarity dimension.

• Honam Mathematical Journal
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• v.34 no.2
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• pp.219-230
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• 2012

In the paper, a symbolic construction is considered to define generalized Cantor-like sets. Lower and upper bounds for the correlation dimension of the sets with a regular condition are obtained with respect to a probability Borel measure. Especially, for some special cases of the sets, the exact formulas of the correlation dimension are established and we show that the correlation dimension and the Hausdorff dimension of some of them are the same. Finally, we find a condition which guarantees the positive correlation dimension of the generalized Cantor-like sets.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.733-738
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• 2008

Using an information of dimensions of divergence points, we give full information of dimensions of the completely decomposed class of the lower(upper) distribution sets of a self-similar Cantor set. Further using a relationship between the distribution sets and the subsets generated by the lower(upper) local dimensions of a self-similar measure, we give full information of dimensions of the subsets by the local dimensions.

• Korean Journal of Mathematics
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• v.16 no.2
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• pp.157-163
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• 2008

Using informations of subsets of divergence points and the relation between members of spectral classes, we give another complete decomposition of spectral classes generated by lower(upper) local dimensions of a self-similar measure on a self-similar Cantor set with full information of their dimensions. We note that it is a complete refinement of the earlier complete decomposition of the spectral classes. Further we study the packing dimension of some uncountable union of distribution sets.

• Korean Journal of Mathematics
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• v.12 no.1
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• pp.1-5
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• 2004

We study Hausdorff and packing dimensions of subsets of a coding space with an ultra metric from a multifractal spectrum induced by a self-similar measure on a Cantor set using a function satisfying a H$\ddot{o}$lder condition.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.4
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• pp.357-364
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• 2006

We study Hausdorff and packing dimensions of subsets of a general coding space with a generalized ultra metric from a multifractal spectrum induced by a self-similar measure on a self-similar Cantor set using a function satisfying a H${\ddot{o}}$older condition.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.19 no.37
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• pp.233-240
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• 1996

Julia set, Fatou set와 Mandelbrot set 가 컴퓨터에 의하여 도형화된 후부터 혼돈 역학체계 (chaotic dynamical system)에 대한 연구가 모든 학계에 비상한 관심을 모으고 있으며 특히 수학자들에 의하여 많은 연구가 이루어지고 있다. 또한 혼돈 역학체계를 기초로 하여 컴퓨터 그래픽스를 이용한 후랙탈(fractal)들의 매혹적인 시각적 표현으로 인하여 최근들어 과학자들 뿐 아니라 일반대중의 후랙탈에 대한 관심이 매우 높아지고 있다. 후랙탈이란 말은 라틴어 fractus(부서진 상태를 뜻함)에서 유래되었으며 1975년 Mandelbrot가 수학 및 자연계의 비정규적 패턴들에 대한 체계적 고찰을 담은 자신의 에세이의 표제를 주기 위해서 만들었다(〔6〕). 후랙탈을 기술하는데 있어서 가장 중요한 양은 차원(dimension)으로, 예컨데 Cantor 1/3 집합은 길이 1인 선분으로부터 시작하야 매 단계마다 모든 선분들의 가운데 1/3을 잘라내는 것을 무한히 반복함으로써 얻어지는데 이 집합의 Lebesgue measure는 0이지만 후랙탈 차원은 log2/log3 로 정수차원이 아닌 실수차원을 갖으며 또한 Cantor 1/3집합은 연속이 아니면서 점도 선도 아닌 집합인 것이다. 이 논문에서는 Box counting dimension 과 Hausdorff dimension에 대한 몇 가지 정의를 하고 정리 2.6, 정리2.7 및 정리 3.3을 증명함으로써 어떤 성질을 갖는 후랙탈의 가장 중요한 양인 후랙탈 차원에 대하여 논의 하고자 한다.