On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm

In this paper, a novel escape-time algorithm is proposed to calculate the connectivity's degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.

Automated summary

In this paper, a novel escape-time algorithm is proposed to calculate the connectivity's degree of Julia sets generated from polynomial maps. The algorithm combines quantitative analysis and visual display to measure the connectivity of Julia sets, providing a connectivity criterion method with an output value Co within the range of [0,1]. By modifying the classical escape-time algorithm, the Julia sets are drawn into different brightnesses according to different Co values, where darker colors indicate better connectivity. The efficiency of the algorithm is assessed through numerical results.