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Find all the entire functions $f$ such that $$f(\sin z)=\sin(f(z)),\quad z\in\mathbb C.\tag {*}$$

The motivation to ask this question is an old post Find all real polynomials $p(x)$ that satisfy $\sin(p(x))=p(\sin(x))$. In the post, the proof is based on the real background, such as the equation $\sin z=z$ has a unique real root $x=0$, but in the complex field, there are infinite roots. It is easy to check that $$f\equiv0, \quad f(z)=\pm z,\quad f(z)=\pm\sin z,\pm\sin (\sin z),\cdots$$ satisfy the equation $(*)$. Are these all the entire function satisfying equation (*).

My thought is to expand $f$ to its Taylor series $$f(z)=\sum_{n=0}^{\infty}a_nz^n,\quad \sin z=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}z^n.$$ But no way to advance the next step.

Any comments and hints will welcome! Thanks a lot!

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    $\begingroup$ $f(z)=\sin z$ satisfies that equation too $\endgroup$
    – user8268
    Commented May 7 at 9:54
  • $\begingroup$ @ user8268 You are right, I add it! $\endgroup$
    – Riemann
    Commented May 7 at 9:57
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    $\begingroup$ And $\sin (\sin z)$, $\sin (\sin ( (\sin z)))$ etc. $\endgroup$
    – geetha290krm
    Commented May 7 at 10:00
  • $\begingroup$ @geetha290krm Nice, thank you! $\endgroup$
    – Riemann
    Commented May 7 at 10:01
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    $\begingroup$ for any fixed point $\sin a =a$ the constant function $a$ satisfies the above; this subject is called permutability and there are many results; for $\sin z$ it is known that a permutable $g$ (nonconstant and not $\pm z$) of finite order must be $\pm \sin z$ (and that the number of permutable $g$ is countable) but not much is known about possibilities in infinite order except that they satisfy algebraic ODE's; for $\sin z +p(z), p$ nonconstant polynomial, the permutable functions are indeed iterates (or simple combinations of such) but $\sin$ is exceptional in a well-defined sense $\endgroup$
    – Conrad
    Commented May 7 at 17:46

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