the inconsistency depending on fixpoint-selection

[bo198214]

The speculation behind this is, to extract a functional relation between two u, say u and v, or more precise u_k and v_j, having the same NET b,

As far as I know there is no closed formula for say the function of the upper fixed point for the argument of the lower fixed point. However we can draw a graph, \( f : (1,e)\to(e,\infty) \) though this can extended to \( f : (1,\infty)\to(1,\infty) \) if we choose always the opposite fixed point, i.e. for the fixed point greater than \( e \) we compute the other fixed point that is below \( e \), the graph looks like:
   

Is it -under this functional relation- *necessary*, that the powerseries, constructed by u_k and v_j give different results? Or can it be shown, that they must give the same result?

With some more analytical matrix-operations and numerical checks I now tend to the conclusion, that Asimov's proposal may be proven to be false.

Hey Gottfried, it is a quite well known result, that in most cases the regular iteration of an analytic function at different fixed points gives different (usually only slightly differing) functions. Ecalle referred me even to an article about this phenomenon

"Etude theorique et numerique de la fonction de Karlin-McGregor",
Serge Dubuc, Journal d'Analyse Math.. Vol. 42 (1982 / 83)

and I checked it numerically in the bummer thread.