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Given the following two types of functions $$\begin{aligned} f_1(x)=\alpha x^2-x+\alpha^{-1} \\ f_2(x)=\frac{\alpha\beta x^2-x+1/[\beta(\alpha+1)]}{1-\beta x} \end{aligned}$$ are there any well understood functions $g$ satisfying $f_i(g(x))=g(\gamma x)$ for either $i=1$ or $2$ where $\gamma$ is a constant? (Really a pair of well understood functions $g,h$ such that $f_i(g(x))=g(h(x))$ works).

I realize that this is vague but the goal of this is to find a function $g$ such that $f^n\circ g$ has a simple closed form, where $$f^n=\underbrace{f\circ f\circ\dots\circ f}_{n\text{ times}}$$ I've included my general interest so that if anything comes to mind people can answer for either, but for the point of narrowing down the question, what solutions are there to the functional equation $$g(x)^2-g(x)+1=g(\gamma x)$$ for some constant $\gamma$? This of course is the consideration with $i=1$ and taking $\alpha=1$ in the general form of $f_1$.

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