05/22/2008, 09:22 PM
This must mean that:
\( \text{ilog}(f^{\circ t}) = t \text{ilog}(f) \)
so
\( f^{\circ t} = \text{ilog}^{-1}(t \text{ilog}(f)) \)
At first I thought this was either an Abel or Schroeder function, but it seems that it is neither. Because I've also heard Abel functions referred to as "iterational logarithm" before (I think Peter Walker calls them this). So how is this related to Abel functions? What is this? Since it cannot be uniquely identified by the term "iterational logarithm", then can it be called a Hadamard function? a Jabotinsky function?
I am having a lot of trouble comparing this to Abel and Schroeder functions. The best I can do is solve all of them for n:
\( n = \frac{\text{ilog}(f^n(x))}{\text{ilog}(x)}
= \log_c\left(\frac{\sigma(f^n(x))}{\sigma(x)}\right) = \alpha(f^n(x)) - \alpha(x) \)
Andrew Robbins
\( \text{ilog}(f^{\circ t}) = t \text{ilog}(f) \)
so
\( f^{\circ t} = \text{ilog}^{-1}(t \text{ilog}(f)) \)
At first I thought this was either an Abel or Schroeder function, but it seems that it is neither. Because I've also heard Abel functions referred to as "iterational logarithm" before (I think Peter Walker calls them this). So how is this related to Abel functions? What is this? Since it cannot be uniquely identified by the term "iterational logarithm", then can it be called a Hadamard function? a Jabotinsky function?
I am having a lot of trouble comparing this to Abel and Schroeder functions. The best I can do is solve all of them for n:
\( n = \frac{\text{ilog}(f^n(x))}{\text{ilog}(x)}
= \log_c\left(\frac{\sigma(f^n(x))}{\sigma(x)}\right) = \alpha(f^n(x)) - \alpha(x) \)
Andrew Robbins
