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

[Ivars]

\( f(x) = \ln(x) \text{ if } x>0 \)
\( f(x) = \ln(-x) \text{ if }x<0 \)

As Henryk noted, this can be easier expressed as \( f(x)=\ln(|x|) \)

I wonder what the continuous iteration including negative of this function would mean?

\( f^{\circ t}(x) \) for real t ?

Would it mean that Integral over t from

Integral dt \( {f^{\circ t}(x)/t}= -\Omega \)?

Integral dt \( {f^{\circ t}(1/x)/t}= \Omega \)?


Ivars