(Further looking at the (non-) periodicity...)
We can write the iteration of b_p more convenient. Remember my convention
With this we can formulate another recursion, simple and based on the angular parameter p: (I swith to tex-notation because of the use of indices)
\(
\hspace{48} p_1 = p*I - \exp(p*I) = -\cos(p) + I*(p - \sin(p)) \\
\hspace{48} p_h = p_1 + \exp(p_{h-1}) \\
\)
where always
\(
\hspace{48} b_h = b^\text{\^\^h } = \exp(\exp(p_h))
\)
The iteration p_h has here the nested form to the depth h:
\(
\hspace{48} p_h = p_1 + \exp(p_1 + \exp(p_1 + \exp(p_1+ ...))))
\)
and it looks very promising, that this can be periodic or even constant only in few and possible exotic cases. Surely, that does not automatically mean, that \( b^\text{\^\^h }= \exp(\exp(p_h)) \) cannot be periodic, as the example of h and sin(h) shows, but it is a strong hint.
Additionally, we can also insert an alternative startexponent z0 as given in my other examples; we have then a small modification:
\(
\hspace{48} p_1 = p*I - \exp(p*I) \\
\hspace{48} p_2 = p_1 + \log(z0) \\
\hspace{48} p_h = p_1 + \exp(p_{h-1}) \text{ // for h>2 } \\
\\
\hspace{24} \text{ where } \\
\hspace{48} b_2 = b^{z_0} \\
\hspace{48} b_h = \exp(\exp(p_h)) = b^{b_{h-1}}
\)
Here it seems interesting, that by the recursion the initial value z0 disappears into the tail (notational) of the p1+exp(p1+exp(...))) - expression.
Also I doubt by this now, that there could be a substantial difference between the irrational and the rational periods - but well, we have the difference between binomials of integer and non-integer parameters, so....
Hmmm....
Gottfried
Added another picture:
![[Image: Period4.png]](http://go.helms-net.de/math/tetdocs/_equator/Period4.png)
(Hmm, now it would be interesting to trace the trajectories backwards...)
We can write the iteration of b_p more convenient. Remember my convention
Code:
.
p = Pi/2 ~ 1.57079632679
u_p = e^(p*I) = cos(p) + I*sin(p) = 1*I
t_p = exp(u) = e^e^(p*I) ~ 0.540302305868 + 0.841470984808*I
b_p = exp(u/t) = e^(e^(p*I - e^(p*I)) ) ~ 1.98933207608 + 1.19328219947*IWith this we can formulate another recursion, simple and based on the angular parameter p: (I swith to tex-notation because of the use of indices)
\(
\hspace{48} p_1 = p*I - \exp(p*I) = -\cos(p) + I*(p - \sin(p)) \\
\hspace{48} p_h = p_1 + \exp(p_{h-1}) \\
\)
where always
\(
\hspace{48} b_h = b^\text{\^\^h } = \exp(\exp(p_h))
\)
The iteration p_h has here the nested form to the depth h:
\(
\hspace{48} p_h = p_1 + \exp(p_1 + \exp(p_1 + \exp(p_1+ ...))))
\)
and it looks very promising, that this can be periodic or even constant only in few and possible exotic cases. Surely, that does not automatically mean, that \( b^\text{\^\^h }= \exp(\exp(p_h)) \) cannot be periodic, as the example of h and sin(h) shows, but it is a strong hint.
Additionally, we can also insert an alternative startexponent z0 as given in my other examples; we have then a small modification:
\(
\hspace{48} p_1 = p*I - \exp(p*I) \\
\hspace{48} p_2 = p_1 + \log(z0) \\
\hspace{48} p_h = p_1 + \exp(p_{h-1}) \text{ // for h>2 } \\
\\
\hspace{24} \text{ where } \\
\hspace{48} b_2 = b^{z_0} \\
\hspace{48} b_h = \exp(\exp(p_h)) = b^{b_{h-1}}
\)
Here it seems interesting, that by the recursion the initial value z0 disappears into the tail (notational) of the p1+exp(p1+exp(...))) - expression.
Also I doubt by this now, that there could be a substantial difference between the irrational and the rational periods - but well, we have the difference between binomials of integer and non-integer parameters, so....
Hmmm....
Gottfried
Added another picture:
(Hmm, now it would be interesting to trace the trajectories backwards...)
Gottfried Helms, Kassel
