Superior Cantor Sets and Superior Devil Staircases

Mandelbrot, in 1975, coined the term fractal and included Cantor set as a classical example of fractals. The Cantor set has wide applications in real world problems from strange attractors of nonlinear dynamical systems to the distribution of galaxies in the universe (Schroder, 1990). In this article, we obtain superior Cantor sets and present them graphically by superior devil’s staircases. Further, based on their method of generation, we put them into two categories. DOI: 10.4018/jalr.2010102106 IGI PUBLISHING This paper appears in the publication, International Journal of Artificial Life Research, Volume 1, Issue 1 edited by E. Stanley Lee and Ping-Teng Chang © 2010, IGI Global 701 E. Chocolate venue, Hersh y PA 17033-1240, USA Tel: 717/533-8845; Fax 717/533-8661; URL-http://www.igi-global.com ITJ 5520 International Journal of Artificial Life Research, 1(1), 78-84, January-March 2010 79 Copyright © 2010, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. Recently, Rani, and Kumar (2004a, 2004b) introduced superior iterations in the study of fractals and chaos and described this powerful tool to improve various geometric objects effectively in a series of papers (Kumar et al., 2005; Negi et al., 2008; Rani et al., 2005, 2008, 2009). In this article, inspired by the success of superior iterations in the study of fractals and its applications, we generate superior Cantor sets. Further, we present them graphically via superior Devil’s staircases. The work embodied in this article is organized in the following manner. In Section 2, we give some preliminaries to be used in the sequel. The intent of Section 3 is to obtain superior Cantor sets and generate superior devil’s staircases by running computer programs. Conclusion of the work is given in Section 4, while the algorithm used to generate a superior Devil’s staircase is given in the last section.