# Fractal property of generalized M-set with rational number exponent

Liu, Shuai and Cheng, Xiaochun and Lan, Caihe and Fu, Weina and Zhou, Jiantao and Li, Qianzhong and Gao, Guanglai
(2013)
*Fractal property of generalized M-set with rational number exponent.*
Applied Mathematics and Computation, 220
.
pp. 668-675.
ISSN 0096-3003

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## Abstract

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.

Item Type: | Article |
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Research Areas: | A. > School of Science and Technology > Computer and Communications Engineering A. > School of Science and Technology > Computer Science > Artificial Intelligence group |

Item ID: | 12731 |

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Depositing User: | Users 3197 not found. |

Date Deposited: | 22 Nov 2013 12:15 |

Last Modified: | 13 Oct 2016 14:29 |

URI: | http://eprints.mdx.ac.uk/id/eprint/12731 |

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