Fractal property of generalized M-set with rational number exponent

摘要

Dynamic systems described by fc(z) = z2 + c is called Mandelbrot set (M-set), which is important for fractal and chaos theories due to its simple expression and complex structure. fc(z) = zk + 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 |p–q| isomorphic parts.