| 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 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>o</mi></mrow></semantics></math></inline-formula> in the range of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. The smaller the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>o</mi></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>o</mi></mrow></semantics></math></inline-formula> values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.
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