Dyadic Cantor set and its kinetic and stochastic counterpart

Firstly, we propose and investigate a dyadic Cantor set (DCS) and its kinetic counterpart where a generator divides an interval into two equal parts and removes one with probability \((1-p)\). The generator is then applied at each step to all the existing intervals in the case of DCS and to only one...

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Bibliographic Details
Published inarXiv.org
Main Authors Hassan, M K, Pavel, N I, Pandit, R K, Kurths, J
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 01.01.2014
Subjects
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Distribution functions
Fractal analysis
Fractal geometry
Fractals
Mathematical models
Physics - Statistical Mechanics
Probability theory
Size distribution
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Summary:Firstly, we propose and investigate a dyadic Cantor set (DCS) and its kinetic counterpart where a generator divides an interval into two equal parts and removes one with probability \((1-p)\). The generator is then applied at each step to all the existing intervals in the case of DCS and to only one interval, picked with probability according to interval size, in the case of kinetic DCS. Secondly, we propose a stochastic DCS in which, unlike the kinetic DCS, the generator divides an interval randomly instead of equally into two parts. Finally, the models are solved analytically; an exact expression for fractal dimension in each case is presented and the relationship between fractal dimension and the corresponding conserved quantity is pointed out. Besides, we show that the interval size distribution function in both variants of DCS exhibits dynamic scaling and we verify it numerically using the idea of data-collapse.
ISSN:2331-8422
DOI:10.48550/arxiv.1401.0249