Cantor Set as a Fractal and Its Application in Detecting Chaotic Nature of Piecewise Linear Maps

• Published in: National Academy of Sciences, India. Proceedings. Section A. Physical Sciences India. Proceedings. Section A. Physical Sciences, 2020-12, Vol.90 (5), p.749-759
• Main Authors: Choudhury, Gautam, Mahanta, Arun, Sarmah, Hemanta Kr, Paul, Ranu
• Format: Article
• Language: English
• Subjects: Applied and Technical Physics; Atomic; Fractal geometry; Fractals; Iterative methods; Molecular; Optical and Plasma Physics; Physics; Physics and Astronomy; Quantum Physics; Research Article; Topology
• ISSN: 0369-8203; 2250-1762
• DOI: 10.1007/s40010-019-00613-8

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 application in detecting chaotic nature of the dynamics produced by iteration of piecewise linear maps.