Let $a > 1$ and $a≠e$

For the recursive sequence

$a_{n+1}=\frac{\ln \left(a_n^2\right)}{2}$

(In other words recursively applying the operation $\frac{\ln \left(a^2\right)}{2}$)

The values I get appear to be chaotic, however taking the mean of all these values does seem to produce a convergence to $-Ω$ where $Ω=W(1)$

For example for $a=2$ and taking the mean of 67108864 iterations, the value I get is $-0.5690610554745104$ which is $-Ω$ to 2 decimal places, however this inaccuracy may be in part due to the level of accuracy of the python program I used which would have rounded to a number of decimal places at each iteration.

I am aware that if the original sequence were to converge, it would converge to the solution of $ln(−x)=x$ which is indeed $-Ω$ , however the original sequence does not converge, so my question is why does the mean of all the numbers in the chaotic sequence converge to this value, despite it appearing to be chaotic?

  • $\begingroup$ You might be interested in reading about Césaro series en.wikipedia.org/wiki/Ces%C3%A0ro_mean $\endgroup$ – user159517 Sep 21 '16 at 18:37
  • $\begingroup$ I have looked at them, but I'm not sure, since the operation is recursive, that the mean can be shown by a Césaro series in the form $\frac{1}{n}\sum _{i=1}^na_i$ $\endgroup$ – Cubbs Sep 21 '16 at 18:41

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