Today I reviewed this thread and thought I could add some more info here.

First, I'd say, that the base b0~1.71290*I as suggested by Ioannis (Galidakis) in the thread sci.math (200 indeed provides an orbit which is an "aequator". I call it aequator since it seems, that it doesn't diverge to infinity nor to a triple of separated fixpoints nor to one fixpoint.(Compare the first pictures in this thread which show the behave of bases greater and smaller than b0) This base b0 is thus one other interesting complex constant in the study of iteration of exponentials; perhaps we may call it "aequator-constant".

The computation due to Ioannis (translated to Pari/GP):

That this base provides an aequator is also backed by another new observation: if the initial point x0, from where the iteration starts, is nearer to the fixpoint, say x0=0.5 then we still get an aequator-line, and even more smooth - the orbit approximates a circle around the fixpoint: without approximating or diverging. Here are pictures for some other starting points of iteration. (The base is always the same).

The complex point x0=1.0+1.0*I was taken:

The complex point x0=0.9+0.9*I was taken:

The complex point x0=0.7+0.7*I was taken:

The complex point x0=0.5+0.5*I was taken:

Now it was interesting, whether the original line, x0=0 is also an equator. I computed a lot of iterates, but the empirical result is inconclusive. There are three critical places, where (for higher bases) the entries to the divergence to the triple of oscillating fixpoints reside. I even guess, that we get something like a snowflake-curve, not smooth like with the other startingpoints x0.

The complex point x0=0.0+0.0*I (the "original"/reference) was taken:

Here I took two computations (and two colors) for the plot because of need of lots of tons of iterates to get a clue of the shape in the sparse regions. The blue points are the first about two-thousand iterates, with an implicte float-accuracy of 400 decimal digits. We see, that there are very sparse regions. After that I switched to 800 and 1200 digits precision and went up to 80000 iterates - no clue, whether the errors accumulate to something horrible. From that 80000 iterates I deleted all in the inner area so I took only the points in the sparse areas into the plot (red dots). It seems, the values are not completely messed, since in principle the blue and red points join reasonably to one curve.

Hmm... what does this tell me?

For instance: is that aequator through x0=0 asymptotically dense? Is it indeed an aequator at all? Is it enclosed in some disk?

For another instance: what does this mean in regard to fractional iteration? Assume an orbit with a starting value x0 inside the limiting aequator (through x0=0), say through x0=0.5+0.5*I which seems to fill a smooth, closed line densely. Are also all fractional iterates on that curve?

Gottfried

First, I'd say, that the base b0~1.71290*I as suggested by Ioannis (Galidakis) in the thread sci.math (200 indeed provides an orbit which is an "aequator". I call it aequator since it seems, that it doesn't diverge to infinity nor to a triple of separated fixpoints nor to one fixpoint.(Compare the first pictures in this thread which show the behave of bases greater and smaller than b0) This base b0 is thus one other interesting complex constant in the study of iteration of exponentials; perhaps we may call it "aequator-constant".

The computation due to Ioannis (translated to Pari/GP):

Code:

`b=I*solve(x=1,2,abs(LW(-log(b*I)))-1) \\LW is the lambert-W-function (implemented using wikipedia-code)`

\\ b=1.7129360403744179818*I

t = h(b) \\ 0.3920635 + 0.4571543*I by h()-function

x=0

data = vectorv(3*210,r, x=b^x);

That this base provides an aequator is also backed by another new observation: if the initial point x0, from where the iteration starts, is nearer to the fixpoint, say x0=0.5 then we still get an aequator-line, and even more smooth - the orbit approximates a circle around the fixpoint: without approximating or diverging. Here are pictures for some other starting points of iteration. (The base is always the same).

The complex point x0=1.0+1.0*I was taken:

The complex point x0=0.9+0.9*I was taken:

The complex point x0=0.7+0.7*I was taken:

The complex point x0=0.5+0.5*I was taken:

Now it was interesting, whether the original line, x0=0 is also an equator. I computed a lot of iterates, but the empirical result is inconclusive. There are three critical places, where (for higher bases) the entries to the divergence to the triple of oscillating fixpoints reside. I even guess, that we get something like a snowflake-curve, not smooth like with the other startingpoints x0.

The complex point x0=0.0+0.0*I (the "original"/reference) was taken:

Here I took two computations (and two colors) for the plot because of need of lots of tons of iterates to get a clue of the shape in the sparse regions. The blue points are the first about two-thousand iterates, with an implicte float-accuracy of 400 decimal digits. We see, that there are very sparse regions. After that I switched to 800 and 1200 digits precision and went up to 80000 iterates - no clue, whether the errors accumulate to something horrible. From that 80000 iterates I deleted all in the inner area so I took only the points in the sparse areas into the plot (red dots). It seems, the values are not completely messed, since in principle the blue and red points join reasonably to one curve.

Hmm... what does this tell me?

For instance: is that aequator through x0=0 asymptotically dense? Is it indeed an aequator at all? Is it enclosed in some disk?

For another instance: what does this mean in regard to fractional iteration? Assume an orbit with a starting value x0 inside the limiting aequator (through x0=0), say through x0=0.5+0.5*I which seems to fill a smooth, closed line densely. Are also all fractional iterates on that curve?

Gottfried

Gottfried Helms, Kassel