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 ...