Pre-note: Sometimes I'm considering ideas, which may be a bit far out, strange or even trivial but not been resolved in a second view (well maybe they would on a third view...), but if they seem(ed) surprising and interesting enough I posted some of them here already. I think in general this is appropriate, but such sketchy things may marked as such, so the tag "UFO" for such msgs may be the most meaningful.

Feel free to just ignore "UFO"-tagged msgs ...

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I looked for a list of the complex fixpoints of the tetration and whether there are some more interesting regularities. So, for instance the base b=sqrt(2): let's look at the list of the fixpoints.

First: as well known, there is only one attracting fixpoint a=2.

Then we have the repelling fixpoint r0=4.

And then the infinite list of complex fixpoints, all repelling.

Ok, if we apply the iterated log instead of the iterated exponential they become attracting and the fixpoint a0 becomes repelling.

How do we get the different fixpoints r0,r1,r2, ... r_k,... (with each r_k also the conjugate is a fixpoint) ? We use the k'th branch of the log, so

r_k = lim_{n->inf} x = (log(x) + k*2*Pi*I ) / log(b)

When I looked at the plot of that (isolated) fixpoints I asked, whether we could interpolate a line. And indeed, we can simply take any real k to get

r_k = lim_{n->inf} x = (log(x) + k*2*Pi*I ) / log(b)

converging!

So this means, we have in fact an attracting continuous curve, and the usual fixpoints are simply that where k is integer...

I've drawn a plot with a couple of bases:

b = exp(u*exp(-u)) , where if u=log(2) we have our favorite one b=sqrt(2). We get the first two fixpoints

r0 = 4 , r1 ~ 9.09... + i*21.5 etc;

and the fixpoint a = 2, which is repelling if we use logarithmizing.

The plot shows the curves for u = 0.5, u=log(2), u=0.85, u=0.95, where dots are inserted at the integer k's (marking the commonly used fixpoints).

It is interesting to approach the critical base b=e^(1/e), meaning u->1 from below, however I do not yet have a good idea of the geometry of that limiting curves.

Here are the original plot, the value of the imaginary part scaled by asinh() (which means nearly a log-scaling, but allows the zero- and negativ values to be rescaled) and two details near the real fixpoints to see the smoothness of the curves.

Gottfried

Feel free to just ignore "UFO"-tagged msgs ...

------------------

I looked for a list of the complex fixpoints of the tetration and whether there are some more interesting regularities. So, for instance the base b=sqrt(2): let's look at the list of the fixpoints.

First: as well known, there is only one attracting fixpoint a=2.

Then we have the repelling fixpoint r0=4.

And then the infinite list of complex fixpoints, all repelling.

Ok, if we apply the iterated log instead of the iterated exponential they become attracting and the fixpoint a0 becomes repelling.

How do we get the different fixpoints r0,r1,r2, ... r_k,... (with each r_k also the conjugate is a fixpoint) ? We use the k'th branch of the log, so

r_k = lim_{n->inf} x = (log(x) + k*2*Pi*I ) / log(b)

When I looked at the plot of that (isolated) fixpoints I asked, whether we could interpolate a line. And indeed, we can simply take any real k to get

r_k = lim_{n->inf} x = (log(x) + k*2*Pi*I ) / log(b)

converging!

So this means, we have in fact an attracting continuous curve, and the usual fixpoints are simply that where k is integer...

I've drawn a plot with a couple of bases:

b = exp(u*exp(-u)) , where if u=log(2) we have our favorite one b=sqrt(2). We get the first two fixpoints

r0 = 4 , r1 ~ 9.09... + i*21.5 etc;

and the fixpoint a = 2, which is repelling if we use logarithmizing.

The plot shows the curves for u = 0.5, u=log(2), u=0.85, u=0.95, where dots are inserted at the integer k's (marking the commonly used fixpoints).

It is interesting to approach the critical base b=e^(1/e), meaning u->1 from below, however I do not yet have a good idea of the geometry of that limiting curves.

Here are the original plot, the value of the imaginary part scaled by asinh() (which means nearly a log-scaling, but allows the zero- and negativ values to be rescaled) and two details near the real fixpoints to see the smoothness of the curves.

Gottfried

Gottfried Helms, Kassel