Application of CR Iteration Scheme in the Generation of Mandelbrot Sets of zp+logct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z^p + \\log c^t$$\\end{document} Function

Abstract

Mandelbrot set is one of the most fascinating objects in mathematics, so in the literature, one can find many studies related to this set. Moreover, there are many different extensions and generalizations of the classical Mandelbrot set. The two most popular extensions are the use of various kinds of functions and iterations. In this paper, firstly, we replace the constant c in the classical zp+c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z^p + c$$\\end{document} function used in the definition of Mandelbrot set with logct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log c^t$$\\end{document}, where t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\in \\mathbb {R}$$\\end{document} and t≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\ge 1$$\\end{document}. Secondly, we replace the Picard iteration with the CR iteration scheme. For this combination of function and iteration, we prove the escape criterion for the escape-time algorithm used to generate some images of the proposed sets. Moreover, we study the dependency between the iteration’s parameters and two numerical measures (average number of iterations and generation time). We show that this dependency is complex and non-linear.