continuous linear recurrent relations

For a function $f:\mathbb{R}\rightarrow \mathbb{R}$ denote $f_0(x)=x$, $f_n(x)=f(f_{n-1}(x))$. Assume that $f$ satisfies a functional equation
$$f_n(x)+a_{n-1} f_{n-1}(x)+\dots+a_0 f_0(x)\equiv 0$$
for some constant real coefficients $a_{0},a_1,\dots,a_{n-1}$. Assume also that $f$ is continuous. What are conditions for polynomial $t^n+a_{n-1}t^{n-1}+\dots +a_0$ which allow to conclude that $f$ is linear?

Answer

Attribution
Source : Link , Question Author : Fedor Petrov , Answer Author : Community

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