(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)

where always

The iteration p_h has here the nested form to the depth h:

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 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:

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...)

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*I

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)

where always

The iteration p_h has here the nested form to the depth h:

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 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:

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