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The divisor periodic point of escape-time \(N\) of the Mandelbrot set. (English) Zbl 1114.65365

Summary: This paper offers a rendering method based on the escape-time method to draw the Mandelbrot set in different colors. The rendering method chooses different colors according to the distance and the escape-time N, so it can effectively render the Mandelbrot set, which can show us the 3-D view of the Mandelbrot set. According to amplify the part of the Mandelbrot set drawn by the rendering method, we will find some cirques with one heart like the equipotential lines. The center point \(c_{0}\) of the cirques with one heart whose color will be rendered by black has a period, and only those center points \(c_{0}\), whose periods are the divisors of the escape-time N, will show us the cirques with one heart like the equipotential lines around them. We call those center points \(c_{0}\) the divisor periodic points of the escape-time N.

MSC:

65P20 Numerical chaos
37E05 Dynamical systems involving maps of the interval
37M05 Simulation of dynamical systems
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References:

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