Application of CR iteration scheme in the generation of Mandelbrot sets of \(z^p + \log c^t\) function (Q6635497)

In this work, the authors delve into the generation of Mandelbrot sets using a novel approach. The authors introduce two main modifications to the classical Mandelbrot set definition: replacing the constant \(c\) in the function \(z^p+\log c^t\), where \(t\in\mathbb{R}\) and \(t \geq 1\), and replacing the standard Picard iteration with the CR iteration scheme. The paper proves the escape criterion for the escape-time algorithm, which is essential for generating images of the proposed Mandelbrot sets. Additionally, the authors study the dependency between the iteration's parameters and two numerical measures: the average number of iterations and generation time, demonstrating that this relationship is complex and non-linear.\N\NFirstly, the replacement of the constant \(c\) with \(log c^t\) and the use of the CR iteration scheme instead of the Picard iteration are innovative steps. These modifications allow for the exploration of new types of Mandelbrot sets, expanding the scope of fractal geometry research.\N\NSecondly, the authors provide a rigorous proof for the escape criterion for the CR iteration scheme. This proof is crucial for the practical implementation of the escape-time algorithm, enabling the generation of visually appealing and mathematically significant Mandelbrot set images.\N\NThirdly, the study of the dependency between iteration parameters and numerical measures (average number of iterations and generation time) offers valuable insights into the computational aspects of generating Mandelbrot sets. This analysis can guide future research on optimizing fractal generation algorithms.\N\NFinally, the manuscript is well-organized, with clear sections dedicated to definitions, proofs, graphical examples, and numerical simulations. The introduction provides a comprehensive overview of the Mandelbrot set and its various extensions, setting the stage for the authors' contributions.\N\NIn conclusion, this work is a mathematically valuable and practically significant research paper. Therefore, I highly recommend this article to researchers in related fields.