Infinite-corner-point fractal image generation by Newton's method for solving [formula omitted

Abstract

Newton's method is sensitive to an initial guess. It exhibits chaotic behavior that generates interesting fractal images. Most of them have either a finite number of attractors (attractive fixed points) or unbounded Julia sets. In this paper, we show that Newton's method for a family of equations exp −α ζ+z ζ−z −1=0 (for α > 0 and | ζ| = 1) has infinitely many attractors and a bounded Julia set. The dynamics of Newton's method for finding their roots are also visualized.