Numeric characteristics of generalized M-set with its asymptote

The Mandelbrot set (M-set) formulated by f(c)(z) = z(2) + c is important in chaos theory. It has been extended into generalized M set (k-M set) f(c)(z) = z(k) + c by using different exponent k. The escape time algorithm is used to draw fractals, which need preset thresholds and a maximum iteration times as input. This paper presents and proves the upper and lower bounds of generalized M set thresholds. A new method is also introduced to draw k-M set when k is a rational number. This method uses asymptotes of k-M set to approach k-M set from the outside. We prove that k-M set is the limit of these asymptotes. Finally, some experiments are presented to validate our methods. (C) 2014 Elsevier Inc. All rights reserved.