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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?

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