Here ist a picture which shows the mapping of the small positive part of the imaginary axis in heights of h=1/20 when computed by the "polynomial method" with the diagonalization of the 64x64 Carlemanmatrix.

The coordinates of the initial linesegment are computed by the 21 coordinates at the imaginary axis between 0 and 0.5 I. To avoid logarithms of zero this is a bit translated by adding 1e-4*I to the coordinates.

Then 20 iterations with height of 1/20 are computed for that whole initial line. This gives the red skewed rectangle in the rough area (0+0I,0+0.5I,1,1+0.5*I)

Then that matrix of coordinates is completely mapped by natural iterations of the powertower with base 4.

The whole segment becomes more and more distorted and shrinks when iterated to the complex fixpoint.

What is especially interesting me are the "eyes" /the whitespace in the inner spiral - and what about of regions of overlap in there if my initial linesegment where longer, say up to 0.8*I. This direction would be related by complex iteration heights - but I've not yet reliable computations of that heights.

[attachment=1016]

The coordinates of the initial linesegment are computed by the 21 coordinates at the imaginary axis between 0 and 0.5 I. To avoid logarithms of zero this is a bit translated by adding 1e-4*I to the coordinates.

Then 20 iterations with height of 1/20 are computed for that whole initial line. This gives the red skewed rectangle in the rough area (0+0I,0+0.5I,1,1+0.5*I)

Then that matrix of coordinates is completely mapped by natural iterations of the powertower with base 4.

The whole segment becomes more and more distorted and shrinks when iterated to the complex fixpoint.

What is especially interesting me are the "eyes" /the whitespace in the inner spiral - and what about of regions of overlap in there if my initial linesegment where longer, say up to 0.8*I. This direction would be related by complex iteration heights - but I've not yet reliable computations of that heights.

[attachment=1016]

Gottfried Helms, Kassel