05/24/2008, 08:35 AM
bo198214 Wrote:The Abel function counts the iterations of \( f \):
\( \phi(f(f(x)))=\phi(f(x))+1=f(x)+2 \)
\( \phi(f(f(f(x))))=\phi(f(f(x)))+1=f(x)+2+1=f(x)+3 \)
\( \phi(f^{\circ n}(x))=\phi(x)+n \).
The fact that Abel function extends COUNTING to non-integer , negative(?) and complex values of COUNTING is very interesting concept besides the fact it is useful in computations and analysis of iterates.
\( \phi(f^{\circ t}(x))=\phi(x)+t \).
e.g.
\( \phi(f^{\circ I}(x))=\phi(x)+I \).
If we turn it upside down, we can may be use the result of t iterations of a function to define what COUNTING with t (e.g. t=I) means.
We may get functions which are only COUNTABLE in certain ways, so that Abel equation is not possible to solve for all t.
Would that make sense?
Ivars