Topological semiconjugacy preserves topological transitivity Let $(X, f )$, where $X$ is a compact metric space and $f : X → X $ is a continuous function and let $(Y, g)$  where $Y$ is a compact metric space and $g : Y → Y$ is a continuous function. Suppose they are topologically semiconjugate, i.e., there is a continuous surjection $h : X → Y$ such that $h ◦ f = g ◦ h$, then show that the transitivity of $f$ implies the transitivity of $g$.
Definition of topological transitivity: Let $X$ be a metric space and $f : X → X$ continuous. $f$ is said to be topologically transitive if for every pair of nonempty open sets $U$ and $V$ in $X$, there is a positive integer $n$ such that $f^n(U) ∩ V ≠ ∅$.
I really don't know how to start this demonstration, a little help would be much appreciated. Thank you in advance for your answers! 

Hint : $f$ is topologically transitive if and only if there exists a dense orbit.
