Consider the analytic function $g(z)=-z(1-z)$ which is a generator of Logistic Sequence with multiplier $-1$, having 2 global fixed point, $g(0)=0$ and $g(2)=2$.
From classical dynamics, whenever a multiplier, or the derivative $\lambda=g'(L)$ at the fixed point $L$ satisfies $|\lambda|\ne0,1$, then the function's iterations of all complex orders can be constructed. In this case, we have $g'(2)=3$, hence the iterations $g^t$ is constructable.
However, dynamics concern about whether the functional iterations are constructable from other cases. It is proved that $\lambda=0,1,e^{2\pi{i}r}$ where $r$ is irrational are all "constructable". This leads to the question about the case that $\lambda$ is a root of unity: $(g'(0)=-1)$
Let $f$ be a multivalued analytic function, each branch cut of its is a meromorphic function, satisfying $\forall{z}\in\mathbb{C},f(f(z))=g(z)=-z(1-z)$, here the functional composition means for some integers $i,j$, there exists two branch cut of $f$: $f_i,f_j$, such that $f_i(f_j(z))=g(z)$.
To only consider the "$\lambda$ is a root of unity" case, we should avoid using any information about the fixed point $2$ during the construction.
So the question, is whether a multivalued function $f$ exists satisfying all 4 properties below:
1.$$\exists\{i,j\}, f_i(f_j(z))=-z(1-z)$$
2.$$(f(0)=0\text{ or }2\text{ or }f \text{ has a pole at z=0})\land(f(2)\ne2\lor(f(2)=2,f'(2)^2\ne3))$$
3.As $z\to\infty$, $$(f(z)\sim{z^{\sqrt{2}}})\lor(f(z)\sim{z^{-\sqrt{2}}})$$ 4.If $f'(z)$ is analytic, then $$\lim_{z\to0}f'(z)^2=-1$$ And if it exists, how to compute?
An equivalent form of this question is,
Given $$\begin{matrix}h(z)=z-2z^3+z^4\\\alpha^{-1}_0\{h\}(z)=\frac{1}{2} \sqrt{\frac{1}{z}}\left(\frac{\frac{5}{64}-\frac{11\log(z)}{64}}{z^{3/2}}+\frac{1}{4\sqrt{z}}-\frac{11\log(z)}{32 z}+1+o(\frac{1}{z^2})\right)\\\alpha^{-1}_1\{h\}(z)=\lim_{n\to\infty}h^{-n}(\alpha^{-1}_0\{h\}(z+n))\\\alpha^{-1}\{h\}(z)=\alpha^{-1}_1\{h\}(z+C)\text{ where }C\text{ is defined by }\alpha^{-1}_1\{h\}(C)=\frac{1}{2}\end{matrix}$$ satisfying $$\alpha^{-1}\{h\} (z-1)=h^{-1}(\alpha^{-1}\{h\}(z))$$ We ask for such a function $P$ satisfying:
1.$$\forall{z}\in\mathbb{C},P(z)=P(z+2),P(0)=1,P(1)=0$$ 2.$$g(z)=-z(1-z),T(z)=P(z)\alpha^{-1}\{h\}\left(\frac{z}{2}\right)+P(z+1)g\left(\alpha^{-1}\{h\}\left(\frac{z-1}{2}\right)\right)$$ such that $$\forall{z}\in\mathbb{C}\setminus{A},T(z+1)=g(T(z))$$ where $A$ is a measureable set in 1 dimension.
Does $P$ exist? If so, how to compute? If not, why?
