Fractal property of generalized M-set with rational number exponent

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Liu, S., Cheng, X., Lan, C., Fu, W., Zhou, J., Li, Q. and Gao, G. 2013. Fractal property of generalized M-set with rational number exponent. Applied Mathematics and Computation. 220, pp. 668-675. https://doi.org/10.1016/j.amc.2013.06.096
Type Article Fractal property of generalized M-set with rational number exponent Liu, S., Cheng, X., Lan, C., Fu, W., Zhou, J., Li, Q. and Gao, G. 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. Fractals; Mandelbrot set; Generalized Mandelbrot set; Bound; Rational exponent Artificial Intelligence group Elsevier Applied Mathematics and Computation 1873-5649 0096-3003 07 Aug 2013 01 Sep 2013 22 Nov 2013 01 Jan 2013 Published 1-s2.0-S0096300313007455-main.pdfLicenseCC BY 4.0 https://doi.org/10.1016/j.amc.2013.06.096 WOS:000324558600066 English

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