On the Dynamics of Laguerre’s Iteration Method for Finding the th Roots of Unity

Figure 7

Several consecutive zooms into two parts of the boundary for . Note that the basin of convergence is not either connected or simply connected. In particular, it appears that the basin of attraction of the roots consists of infinitely many (quasi-) self-similar disconnected sets (zooms on the left) and infinitely many (quasi-) self-similar “holes” corresponding to basins of attraction of the two-cycle (zooms on the right).