An application of the "polynomial method/Diagonalization" (matrix-size 64x64) which was in the instance of the article shown with base and in that article likely asymptotic to the Kneser method.

Here I provide a picture with focus on complex iteration from one real starting point with base . (I've not yet checked against Sheldon's Kneser-implementation ).

The picture shows roughly circles: along the circumferences the iteration-height is purely imaginary; one revolving means where v is the log of the log of the fixpoint

It is still surprising to me that we can proceed from one real point below the fixpoint to some other real point above the fixpoint - which means to avoid/surpass the infinite height-iteration: just by using imaginary heights...

Gottfried

Here I added two more (hopefully instructive) views;

Here the base-point is z0=0

and here is the iteration to one more negative height, where I had to leave out the infinitely distant point z0 (-> - infinity)

Now I've got my matrices for base b=1.44, near eta. What a mess!

I don't have any idea - there is no obvious divergence in the power series with 64 terms. I also took z0=b^b ~ 1.69 as initial point; the curves originating from z0=1 were even more messed up.

This is the picture base b=1.44 z0=1+0î - I've no explanation so far for the messed curves.

<hr>

P.s.: A 3-D picture with colors indicating height, and "isobares"-grid and the fixpoint shown as peak of infinite height were nicer but I do not know how to draw one(which software). If someone else likes to play with this I can provide the coefficients of the diagonalization matrices in Pari/GP-convention: the computation of that matrices is extremely costly (it needed 6000 secs to be computed and more than 3000 digits decimal precision;I chose then 4000 digits) so it might be interesting to get the ready-made numbers by download instead by a new computation.

Here I provide a picture with focus on complex iteration from one real starting point with base . (I've not yet checked against Sheldon's Kneser-implementation ).

The picture shows roughly circles: along the circumferences the iteration-height is purely imaginary; one revolving means where v is the log of the log of the fixpoint

It is still surprising to me that we can proceed from one real point below the fixpoint to some other real point above the fixpoint - which means to avoid/surpass the infinite height-iteration: just by using imaginary heights...

Gottfried

Here I added two more (hopefully instructive) views;

Here the base-point is z0=0

and here is the iteration to one more negative height, where I had to leave out the infinitely distant point z0 (-> - infinity)

Now I've got my matrices for base b=1.44, near eta. What a mess!

I don't have any idea - there is no obvious divergence in the power series with 64 terms. I also took z0=b^b ~ 1.69 as initial point; the curves originating from z0=1 were even more messed up.

This is the picture base b=1.44 z0=1+0î - I've no explanation so far for the messed curves.

<hr>

P.s.: A 3-D picture with colors indicating height, and "isobares"-grid and the fixpoint shown as peak of infinite height were nicer but I do not know how to draw one(which software). If someone else likes to play with this I can provide the coefficients of the diagonalization matrices in Pari/GP-convention: the computation of that matrices is extremely costly (it needed 6000 secs to be computed and more than 3000 digits decimal precision;I chose then 4000 digits) so it might be interesting to get the ready-made numbers by download instead by a new computation.

Gottfried Helms, Kassel