  We investigate combinatorial dynamical systems on simplicial complexes
considered as {\em finite topological spaces}. Such systems arise in a natural
way from sampling dynamics and may be used to reconstruct some features of the
dynamics directly from the sample. We study the homological persistence of {\em
Morse decompositions} of such systems, an important descriptor of the dynamics,
as a tool for validating the reconstruction. Our framework can be viewed as a
step toward extending the classical persistence theory to "vector cloud" data.
We present experimental results on two numerical examples.
