  Dynamic topological logic ($\mathbf{DTL}$) is a trimodal logic designed for
reasoning about dynamic topological systems. It was shown by Fern\'andez-Duque
that the natural set of axioms for $\mathbf{DTL}$ is incomplete, but he
provided a complete axiomatisation in an extended language. In this paper, we
consider dynamic topological logic over scattered spaces, which are topological
spaces where every nonempty subspace has an isolated point. Scattered spaces
appear in the context of computational logic as they provide semantics for
provability and enjoy definable fixed points. We exhibit the first sound and
complete dynamic topological logic in the original trimodal language. In
particular, we show that the version of $\mathbf{DTL}$ based on the class of
scattered spaces is finitely axiomatisable over the original language, and that
the natural axiomatisation is sound and complete.
