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Wild conjecture about 2 fixpoints.
#1
Let f(z) be a real entire function that is strictly increasing on the real line and has exactly 2 fixpoints A,B both being positive reals.
Also f(0)=1 , f ' (A) =/= 0,1 , f ' (B) =/= 0,1.

This f(z) has 2 regular superfunctions ; superf_A(z) and superf_B(z) based on what fixpoint was used.

Conjecture : the functional equations that hold on the other branches of superf_A(z) are identical to the functional equations that hold on the other branches of superf_B(z).

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Note : We assumed the existance of superfunctions , some functions do not even have that ! ( such as polynomials of degree 2 )
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Remark : We can define a half-iterate that agrees on both fixpoints withing the interval [A,B] but it has singularies at A,B.
A similar conjecture can be made about those singularities.
Or about the singularities of its superfunction.

Example of Remark :

f(z) = (5/4)^z.

For A < z < B define :

g(z) = SUM_k cos(2pi k) f^[k](z)

where the sum runs over all the integers k.

Now g(z) = g((5/4)^z)

So if g(w) = 0 with z<w<(5/4)^z then

w = f^[1/2](z).

Now by assuming everything to be analytic ( is g(z) analytic ? is f^[1/2](z) analytic ? ) ...

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this example unfortunately holds a new conjecture about the analyticity of these functions ... or has this been (dis)proved before ?
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...we can consider these singularies at A,B for the functions in the "same way" (i mean the first conjecture made here ) ... or construct the super and consider those singularities in the same way (ditto).



regards

tommy1729
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