Let $x\in\left[0,\frac{\pi}{2}\right]$ and let $a,b\in (0,1]\subset\mathbb R$.
Define the sequence $$ \theta_0 = x \\ \theta_{n+1} = \arcsin\left(a\sin(b \theta_n)\right),\;n\ge 0 $$
If $a=b=1$ the sequence is constant. Otherwise $\theta_n$ is decreasing, and as it goes to zero, to a first order approximation $\theta_{n+1}\approx ab\theta_n$. In either case we can say that $\theta_n (ab)^{-n}$ converges to a finite limit (this may not be trivial, but I'll take it as given).
Define a function $f_{a,b}$ parameterized by $a,b$ as this limit as a function of $x$ $$ f_{a,b}(x) = \lim_{n\to\infty} \theta_n (ab)^{-n} \quad \text{when}~\theta_0=x $$
$a=1\implies\theta_n = b^n\theta_0\;\land\;f_{1,b}(x)=x$.
Also $\lim_{a\to 0} f_{a,b}(x) = \frac{1}{b}\sin bx$ and $\lim_{b\to 0} f_{a,b}(x) = x$.
Is there a simpler functional form for $f_{a,b}$ for $0<a,b<1$, generally or especially for $b=1/2$?
This function could be applied to a solution for this problem and this one using half-angle formulas.
