Generalized Mandelbrot Sets from a Class of Complex Mapping $z \leftarrow e^{i\phi }(\bar {z})^\alpha + c(\alpha < 0)$

Abstract:

      The nature of critical points from a class of complex mapping $z \leftarrow e^{i\phi }(\bar {z})^\alpha + c\{\alpha < 0,\phi \in [0,2\pi)\}$ was analyzed, the definition of the generalized Mandelbrot sets was given, and a series of the generalized Mandelbrot sets for negative real index number were constructed. Using the experimental mathematics method and the theory of analytic function of one complex variable with computer aided drawing, the fractal features and evolution of the generalized Mandelbrot sets were studied. The results show: 1). The geometry structure of the generalized Mandelbrot sets depends on the parameters of \textit{$\alpha $}, $R$ and the following $\phi$; 2). The generalized Mandelbrot sets for integer index number have symmetry and fractal feature; 3). The generalized Mandelbrot sets for decimal index number have discontinuity and collapse, and their evolutions depend on the choice of the principal range of the phase angle.