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Iterations of ln(mod(ln(x^(1/x))) and number with property Left a[4]n=exp (-(a^(n+1))
#1
I have been playing with iterations of



And of course I tried and

is not very interesting, it converges to 2 values depending on integer iteration number. Here are picture of 200 iterations:
   


is more interesting. This iteration converges to 4 values for each x, cyclicaly,and their dependance on n is shifting depending the region x is in the interval ]0:1[. The convergence values are :



Here is what happens:

   


   


When resolution is increased (step decreased) more and more shifts in phase happen in the region which seems to converge to approximately 0,6529204....

   

   


This number has properties:







So its 2nd selfroot is , while its reciprocal 1,531580266.. is 3rd superroot of e.

It has also following properties, at least approximately numerically, so it might be wrong, but interesting:

if we denote it , than:





So:





........





..........

if this is so, what happens if instead of integer n we take x, so that:

maybe:




I hope I used [4Left] correctly. So:

and



If accuracy is not enough, so it is numerically only 3-4 digits, perhaps going for n-th superroot of e would improve situation?

Probably this is old knowledge.

Ivars
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