If $\mathbb{L} =\mathbb{Z}^D$ and $\mathcal{A}$ is a finite set, then $\mathcal{A}^\mathbb{L}$ is a compact space; a cellular automaton (CA) is a continuous transformation $\Phi:\mathcal{A}^\mathbb{L} \longrightarrow\mathcal{A}^\mathbb{L}$ that commutes with all shift maps. A quasisturmian (QS) subshift is a shift-invariant subset obtained by mapping the trajectories of an irrational torus rotation through a partition of the torus. The image of a QS shift under a CA is again QS. We study the topological dynamical properties of CA restricted to QS shifts, and compare them to the properties of CA on the full shift $\mathcal{A}^\mathbb{L}$. We investigate injectivity, surjectivity, transitivity, expansiveness, rigidity, fixed/periodic points and invariant measures. We also study ‘chopping’: how iterating the CA fragments the partition generating the QS shift.