Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ $$Y\;\;\stackrel{g}{\longrightarrow}\;\; Y$$
Assume:
(1) $X$ and $Y$ are real analytic manifolds.
(2) $f$ is a real analytic diffeomorphism.
(3) $\pi$ is a real analytic surjective map.
Question: Is $g$ real analytic? Additional conditions necessary?