Cantor Set as a Fractal and Its Application in Detecting Chaotic Nature of Piecewise Linear Maps
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RESEARCH ARTICLE
Cantor Set as a Fractal and Its Application in Detecting Chaotic Nature of Piecewise Linear Maps Gautam Choudhury1 • Arun Mahanta2 • Hemanta Kr. Sarmah3 • Ranu Paul4
Received: 27 October 2015 / Revised: 16 August 2016 / Accepted: 13 May 2019 The National Academy of Sciences, India 2019
Abstract We have investigated the Cantor set from the perspective of fractals and box-counting dimension. Cantor sets can be constructed geometrically by continuous removal of a portion of the closed unit interval [0, 1] infinitely. The set of points remained in the unit interval after this removal process is over is called the Cantor set. The dimension of such a set is not an integer value. In fact, it has a ‘fractional’ dimension, making it by definition a fractal. The Cantor set is an example of an uncountable set with measure zero and has potential applications in various branches of mathematics such as topology, measure theory, dynamical systems and fractal geometry. In this paper, we have provided three types of generalization of the Cantor set depending on the process of removal. Also, we have discussed some characteristics of the fractal dimensions of these generalized Cantor sets. Further, we have shown its
& Gautam Choudhury [email protected] Arun Mahanta [email protected] Hemanta Kr. Sarmah [email protected] Ranu Paul [email protected] 1
Mathematical Sciences Division, Institute of Advance Study in Science and Technology, Boragaon, Guwahati, Assam, India
2
Department of Mathematics, Kaliabor College, Nagaon, Assam, India
3
Department of Mathematics, Gauhati University, Guwahati, Assam, India
4
Department of Mathematics, Gauhati University, Guwahati, Assam, India
application in detecting chaotic nature of the dynamics produced by iteration of piecewise linear maps. Keywords Cantor set Fractal Box-counting dimension Chaotic map
1 Introduction Cantor set was discovered by German mathematician George Cantor (1845–1918) in the last part of the nineteenth century. In 1883, he wrote a series of papers that contained the first systematic treatment of the point set topology of real line in which he raised some problems and attracted interest of researchers to the field of set theory. One of these is the classical Cantor set which he forwarded as an example of a perfect, nowhere dense subset of R . A considerable stir was created in the mathematical community since the properties of the Cantor set were found to be counterintuitive. For example, it is hard to imagine points in the Cantor set other than the end points of the intervals which are removed in the process of its construction, yet we know that they must exist. In modern terminology, the Cantor set is a classical example of a perfect set of the closed interval ½ 0; 1 that has the same cardinality as real line whose Lebesgue measure is zero. Throughout much of the twentieth century, Cantor set was considered to be a little more than mathematical curiosity. However, the work of Stephen Smale in the 1960s demonstrated that C
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