Well, here are some pictures for the iteration of the cubic.
This is the plot for the first half-iterate from the three cubic roots z0,z1,z2 of Z=1+0.5*I. Just applied four times using z=z^(3^0.5), for the three z0,z1,z2, then z=z^(3^0.5), and so on, just a couple of times. clearly the naive computation of complex z to the squareroot(3) does not provide meaningful graphs - neither for the integer and even less for the spline-interpolated coordinates of a plot. Well, that's true for the half-iterates of z1 and z2, for the half iterates of z0 the plot seems meaningful:
The direct source of such a chaos is the behave of the software to provide the imaginary value for such a complex root as positive or negative value in a restricted range for arg(z) between -Pi..Pi because it is just modular to 2*Pi. Correcting at least for the appropriate phi for the quadrant in which z resides gives that far better looking graph, where we even can insert smaller iteration steps.
but still with some problem, when the iteration reaches the border of a quadrant, the arg()-function for the iterates oscillates and must be corrected. The detail shows an entertaining oscillation at the negative real axis...
Another correction for the imaginary values, when iterated is tried:
and is even better, but still not perfect. At least we see the three roots much clearer and how they do not lay on the same trajectory. But the computation is still not optimal: now the crossing of the real axis creates problems. See the detail:
Thus I improved the computation using integer iteration to some small roots first, say instead of z1 using z1^^-10 as basis for a trajectory, compute the regular fractional iterates for one unit interval, say y1, and then retransform to y^^10. This gives the acceptable trajectories of the next pictures, shown in the following post (the MyBB-software seems to not allow more than 4 or five attachments per post...)
Gottfried
This is the plot for the first half-iterate from the three cubic roots z0,z1,z2 of Z=1+0.5*I. Just applied four times using z=z^(3^0.5), for the three z0,z1,z2, then z=z^(3^0.5), and so on, just a couple of times. clearly the naive computation of complex z to the squareroot(3) does not provide meaningful graphs - neither for the integer and even less for the spline-interpolated coordinates of a plot. Well, that's true for the half-iterates of z1 and z2, for the half iterates of z0 the plot seems meaningful:
The direct source of such a chaos is the behave of the software to provide the imaginary value for such a complex root as positive or negative value in a restricted range for arg(z) between -Pi..Pi because it is just modular to 2*Pi. Correcting at least for the appropriate phi for the quadrant in which z resides gives that far better looking graph, where we even can insert smaller iteration steps.
but still with some problem, when the iteration reaches the border of a quadrant, the arg()-function for the iterates oscillates and must be corrected. The detail shows an entertaining oscillation at the negative real axis...
Another correction for the imaginary values, when iterated is tried:
and is even better, but still not perfect. At least we see the three roots much clearer and how they do not lay on the same trajectory. But the computation is still not optimal: now the crossing of the real axis creates problems. See the detail:
Thus I improved the computation using integer iteration to some small roots first, say instead of z1 using z1^^-10 as basis for a trajectory, compute the regular fractional iterates for one unit interval, say y1, and then retransform to y^^10. This gives the acceptable trajectories of the next pictures, shown in the following post (the MyBB-software seems to not allow more than 4 or five attachments per post...)
Gottfried
Gottfried Helms, Kassel
