Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x

[Ivars]

If we analyize the distribution of Average over 100 points per Iteration (over 5000 iterations) , then distribution in both cases ln(mod(x) and ln(mod(1/x)) is 3-parameter LogNormal with parameters:


Scale parametes \( \mu=0.91478 \),

Shape parameter \( \sigma =0.04851 \),

location parameter \( \gamma=-1.9302 \)

http:// http://en.wikipedia.org/wiki/Log...stribution

While if we analyze distribution of Average per point x]0.01:0.01:0.99[ (10 000 iterations) at 100 points , we obtain the best fit by Johnson SU distribution , second best by log logistic distribution . This of course requires more points to be made decisive, but both are interesting, especially log logistic because of its many connections and relation to Lambert W function used in solutions of some growth models.

But these things I have to read on...As many others.

Ivars