I have a simple question on rational functions, which appear in the context of rational elliptic surfaces. Let $R$ be a rational functions over say $\mathbb C$, such that $R=P/Q$ with $P$ and $Q$ polynomials. We can study functions $f$ with the property that $$ R(f(x))=R(x),$$ i.e. $f$ leaves $R$ invariant.
As an example, consider $$R(x)=\frac{64 \left(4 x^2-3\right)^3}{x^2-1}.$$ Then for $$f(x)=\frac{2 x \left(x-\sqrt{x^2-1}\right)-3}{2 \sqrt{2 x \left(\sqrt{x^2-1}-x\right)+2}},$$ I find that $R(f(x)=R(x)$.
What is the nature of such functions? Can they be found explicitly for given $R$? I would be grateful for any comments.