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Top1. Introduction
In chaos game, out of a few simple rules, one is selected at random and applied to produce a new point. Repeatedly, a random rule is selected over and over again to produce an attractor. Barnsley (2006) used chaos game in function systems, and it gave rise to interesting fractal objects by combining it with iterations (i.e., IFS). The IFS models have been widely studied and used in graphical generations (see, for instance, Nikiel, 1998; Pickover, 1985; Wijk & Saupe, 2004; Zair & Tosan, 1996). With the advent of chaos game in fractal graphics, much work has been done on its theoretical and applicable aspects. For basic work on chaos game, one may refer Barnsley (2006), Peitgen, Jürgen, and Saupe (1992) and, for recent impressive work on chaos game, Bahar (1997), Jeffrey (1992), Gottfried (1991), and Martyn (2002a, 2002b). Jones (1990) has extended Barnsley’s method of chaos game from triangles to regular polygons (Barnsley, 2006).
A fractal fern may be generated by taking different probabilities in chaos game. Duis et al. (1987) studied the origin of superficial lightning burns with the help of fern-shaped mathematical models. Michelitsch (1990) generated fern like structures using trigonometric functions. In 1999, Chang, Chang, Ting, and Wen (1999) proposed a fixed-point-searching algorithm which automatically determines the original size and the coordinates of a fractal fern. Recently, Ahammer and Devaney (2003, 2005) computed the correlation dimension of fractal ferns and influence of noise on the same, while Zodrow (2007) reconstructed interesting tree ferns. Fractal ferns have enjoyed its applications in many areas of mathematical sciences such as biology, chemistry, physics and coding theory. For details, one may refer to Barnsley (1989), Korn (1993), Xu, Leung, Liang, and Leung (2003) and Zhu, Hsu, Zhou, Terrones, Kroto, and Walton (2001).
Methods of function iterations find staggering applications in several areas of mathematical sciences. In particular, theory of discrete dynamics is the study of such iterative processes. In general, an IFS is based on function iterations (also called Picard iterations). Recently, Chandra and Rani (2009) and Rani and Kumar (2004a, 2004b, 2005) introduced superior iterations in the study of fractals and offered impressive comparison in favor of this new host in discrete dynamics. The purpose of this paper is to introduce superior iterations and I-superior iterations as generalized approaches to generation and pattern recognition in fractal ferns. In Section 2, we give preliminaries of chaos game and iterative procedures. In Section 3, we generate and study the pattern of fractal ferns followed by conclusions in Section 4.