(12/06/2012, 12:10 AM)Gottfried Wrote: Obviously this is periodic with \( S(f^{[2+h]}(x))=S(f^{[h]}(x)) \) and also sinusoidal. With that base b=1.3 I find for instance at \( x_0 \sim 0.427734366938 \) that \( S(x_0)=0 \).
After this it is surely natural to assume, that beginning at the first iterate we have also that \( S(x_1)=0 \), but it seems also natural to assume, that then the maxima or the minima of the sinusoidal curve of all \( S(x_0)...S(x_1) \) are at the half-iterates between them.
The problem with this is its an " not even wrong " statement.
Just like one could use ALMOST ANY 1-periodic real function to make a new superexp.
Gottfried assumed that they - the extrema - must be half-iterates ... but half-iterates based upon which method ?
Afterall , all methods agree on what an integer iteration is , but most disagree about what a half-iterate is. Not to mention requirements such as analytic and properties such as multiple fixpoints.
So without properties or uniqueness criteria , it is just a half iterate by its OWN method.
Notice \( s(x)/2 \) also satisfies \( F(b^x-1) = -F(x) \) , so the *half* is a "not even wrong" statement and even its value is a "not even wrong" statement.
The disease seems randomness without proven uniqueness or properties.
notice F(x) = F(A(x)) has the general solution F(x) = B( ...+ C(D(x)) + C(A(D(x))) + ...) For an uncountable amount of functions B,C and D.
In fact all solutions are like that. Working with G(inversesuperA(x)) for a periodic suitable G is just an equivalent form.
Although different viewpoint are intresting and this is an intresting question , I do not see a big solution in this for now.
Since Gottfried's example is about a function with 2 fixpoints it means its a solution that is not both analytic and satisfies the functional equation everywhere -- since such a function does not exist --.
Besides we - if I may say we - are mainly looking for superfunction of functions that do not have 2 distinct real fixpoints but rather no real fixpoints ( such as exp(x)).
Also I do not think summability methods preserve properties such as simultaneously being analytic and satisfying the functional equation ... apart from perhaps a finite few trivial exceptions.
There are many uniqueness criterions made up for tetration so it might be possible to find F(x) = B( ...+ C(D(x)) + C(z^(D(x))) + ...) such that these hold.
Otherwise at the moment I see no potential.
