For the text below, define $f^\infty(x) = \lim_{n\to\infty} f^n(x)$ where $f^n = \underbrace{f \circ \ldots \circ f}_{n}$.

Usually 'continued fraction' means continued fraction in $\mathbb{R}$. For example, consider $f(x) = 1+1/x$ where $x \in \mathbb{R}$. In that case, for all $x > 0$, we have $f^\infty(x) = [1;1,1,1,\ldots] = \Phi = \frac{1+\sqrt{5}}{2}$.

Now consider another $f(x) = 1 + 1/x$ that seems to have the same formula as above, but here $x \in \mathbb{R} \to \mathbb{R}$ instead. It should be imaginable that for some $x$, we have $f^\infty(x) = \hat\Phi$ where $\hat\Phi$ is the constant function that always gives $\Phi$. In other words, we can think of $f^\infty(x)$ as a continued fraction in the function space $\mathbb{R} \to \mathbb{R}$.

Has anybody studied such continued fractions in function spaces? I googled for some obvious keywords such as 'continued fraction in function space', but I didn't seem to find anything about what I described above.

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    $\begingroup$ Continued fractions of functions certainly come up, for example in the theory of orthogonal polynomials/Jacobi matrices. $\endgroup$ – Christian Remling Feb 10 '17 at 1:04
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    $\begingroup$ The relation between continued fractions and limits of iterates that you describe holds only for fractional-linear functions. $\endgroup$ – Alexandre Eremenko Feb 10 '17 at 7:09

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