TITLE: Good closed form approximation for iterates of $x^2+(1-x^2)x$
QUESTION [8 upvotes]: Let $f(x) := x^2+(1-x^2)x$. Is there a nice nontrivial closed form approximation $g_n(x)$ over $[0,1]$ for the $n$-fold composition $f^{\circ n}(x)$? Obviously near $0$ we have that $f^{\circ n}(x) = x+nx^2+...$ but this is not much use to me. Rather than try to pin down what "nice" ought to mean, I'll channel Potter Stewart and just say I (and I'm sure also a respondent) would know it upon sight.
One might be tempted to mumble "solve Schroder's equation" but I don't see how that helps. Nor do I see how computing the Carleman matrix of $f$ helps (but for what it's worth, I believe the matrix elements are $M_{jk} := \sum_{r=0}^j \binom{j}{r} (-1)^{j-r} \binom{r}{k-3j-2r}$). Such tactics are suggested in How would I go about finding a closed form solution for $g(x,n) = f(f(f(...(x))))$, $n$ times?

REPLY [2 votes]: FYI, here is a picture illustrating Count's answer at https://math.stackexchange.com/a/1872168/241. Blue curves are the zeroth through tenth iterates of $f$; red curves are obtained by numerically inverting $F(f^{\circ N}(x)) \approx N + F(x)$ for $F(x) := \log \frac{x}{1-x} + \frac{1}{x}$. I will mention that this is an improvement on the exact fourth-order approximation $f^{\circ N}(x) \approx x+Nx^2+(N^2-2N)x^3+([N-1]^3-2[N-1]^2-3[N-1])x^4$ near $0$, and I imagine an improvement on all exact fixed-order approximations.

On a separate note, I will also mention the Carleman approach I originally mentioned can be pursued along the lines of http://www.sciencedirect.com/science/article/pii/S0022247X98959868.