Semiflows on finite topological spaces

In this paper, we study flows and semiflows defined on any given finite topological $T_0$-space $X$. We show that there exist non-trivial semiflows on $X$, unless $X$ is a minimal finite space. Specifically, non-trivial semiflows exist if and only if $X$ contains down beat points, and a non-trivial semiflow is essentially a strong deformation retraction. As a consequence of this result, we provide a new and concise proof that the only flow that can be defined on $X$ is the trivial flow. Finally, we discuss the number of different semiflows that can be defined on $X$ in terms of down beat points and other special points.