Problem statement: Consider $E_\mu(x)=\mu e^x$, where $0<\mu<1/e$. Show that $E_\mu$ has no periodic points that are not fixed points.
It is in my understanding that what we need to show is that $E^{\circ n}_\mu(\overline{x})=\overline{x}$ for all $n\in\mathbb{N}$, that is we need to show that the orbit of any fixed point contains only one element. In this case, since $E_\mu(x)=e^{x-\varepsilon}$ for any $\varepsilon > 1$, we can find that $\overline{x}=\ln(\overline{x})+\varepsilon$. And now is what seems to be the "foggiest" part: to show that $E^{\circ n}_\mu(\overline{x})=\overline{x}$ for all $n\in\mathbb{N}$, we need to express $E^{\circ n}_\mu(\overline{x})$ explicitly, or use induction in some tricky way.
I'd appreciate some hints and/or clarifications.