On the construction, properties and Hausdorff dimension of random Cantor one pth set

Sudesh Kumari; Renu Chugh; Jinde Cao; Chuangxia Huang; Sudesh Kumari; Renu Chugh; Jinde Cao; Chuangxia Huang

doi:10.3934/math.2020202

AIMS Mathematics

2020, Volume 5, Issue 4: 3138-3155. doi: 10.3934/math.2020202

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Research article

On the construction, properties and Hausdorff dimension of random Cantor one p^th set

1.
Department of Mathematics, Government College for Girls Sector 14, Gurugram 122001, India
2.
Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, India
3.
Research Center for Complex Systems and Network Sciences, School of Mathematics, Southeast University, Nanjing 210096, China
4.
School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha 410114, China

Received: 16 December 2019 Accepted: 16 March 2020 Published: 24 March 2020
MSC : 54H20, 37H99, 60B05, 49K45

In 1883, German Mathematician George Cantor introduced Cantor ternary set which is a self-similar fractal. K. J. Falconer (1990) defined random Cantor set with statistical self-similarity. The purpose of this paper is to introduce generalized random Cantor sets (one 5^th, one 7^th and in general one p^th). Some properties and results of random Cantor one p^th set have also been obtained. We compute Hausdorff dimension of random Cantor one p^th sets and show that Hausdorff dimension of these random Cantor sets is less than that of Hausdorff dimension of Cantor one p^th sets, calculated by Ashish et al. (2013).
- nonlinear dynamics,
- random construction,
- random Cantor one p^th set,
- Hausdorff dimension,
- martingale,
- probability
Citation: Sudesh Kumari, Renu Chugh, Jinde Cao, Chuangxia Huang. On the construction, properties and Hausdorff dimension of random Cantor one pth set[J]. AIMS Mathematics, 2020, 5(4): 3138-3155. doi: 10.3934/math.2020202

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Abstract

In 1883, German Mathematician George Cantor introduced Cantor ternary set which is a self-similar fractal. K. J. Falconer (1990) defined random Cantor set with statistical self-similarity. The purpose of this paper is to introduce generalized random Cantor sets (one 5^th, one 7^th and in general one p^th). Some properties and results of random Cantor one p^th set have also been obtained. We compute Hausdorff dimension of random Cantor one p^th sets and show that Hausdorff dimension of these random Cantor sets is less than that of Hausdorff dimension of Cantor one p^th sets, calculated by Ashish et al. (2013).

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