A dynamical system is called Devaney chaotic is it is (i) transitive, (ii) periodic points are dense, and (iii) the system depends sensitively on initial conditions.
My question is if Devaney chaos is maintained by topological semi-conjugacy.
More precisely:
Let $f\colon X\to X$ and $g\colon Y\to Y$ be continuous and $h\colon Y\to X$ be a continuous surjection such that $h\circ g=f\circ h$. Suppose that $g$ is Devaney chaotic on $Y$. Does it then follow that $f$ is also Devaney chaotic on $X$?