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Let $c$ be any positive real number.

Let $z$ be a complex number.

Let $g(c,z)$ be some locally analytic function that is the $c$ th iteration of the entire function $f(z)$ with $g(0,z)=f(z)$.

Let $a$ be a real number and $f(z)$ a function such that

$1)$ $a$ is not a fixpoint or cyclic point of $f(z)$.

$2)$ For the entire function $f(z)$ and the real $a$ we have that $g(c,z) = a$ always has a solution where $z$ is strictly real. (for any $c$)

What is the name for such an $a$ if any exist at all ?

How to find such $a,f(z)$ and $g(c,z)$ ?

Or when some are given ; find the others or prove that they cannot exist ?

Good free online references are welcome.

Im not sure how to put this into a clear short title ...

  • 0
    What does "the $c$th iteration" mean when $c$ is not an integer?2012-11-10
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    Think of it in terms of locally analytic abel functions and their inverses ... (or like tetration and such).2012-11-10
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    Does it hold that $g(c_1,(g(c_2,z))=g(c_1+c_2+1,z)$ for $c_1,c_2\not\in\mathbb{Z}$?2012-11-10
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    Why the $ +1$ ?2012-11-10
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    I would divide the problem into three cases 1) case where f(z) is real valued, and $f^n(z)$ is real valued for all integers (or at least positive integers). 2) f(z) is not real valued, but for some value of n, $f^n(z)$ is real valued, and for all integers greater than n. 3) all other cases, it would seem there would have to be real values of $\alpha$ for which there is no solution of $f^c(z)=\alpha$, for any particular $g(c,z)$.2012-11-11

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