Fractal property of generalized M-set with rational number exponent

Dynamic systems described by f(c)(z) = z(2) + c is called Mandelbrot set (M-set), which is important for fractal and chaos theories due to its simple expression and complex structure. f(c)(z) = z(k) + c is called generalized M set (k-M set). This paper proposes a new theory to compute the higher and lower bounds of generalized M set while exponent k is rational, and proves relevant properties, such as that generalized M set could cover whole complex number plane when k < 1, and that boundary of generalized M set ranges from complex number plane to circle with radius 1 when k ranges from 1 to infinite large. This paper explores fractal characteristics of generalized M set, such as that the boundary of k-M set is determined by k, when k = p/q, where p and q are irreducible integers, (GCD(p, q) = 1, k > 1), and that k-M set can be divided into vertical bar p-q vertical bar isomorphic parts. (C) 2013 The Authors. Published by Elsevier Inc. All rights reserved.