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We call a continuous self-map that reveals itself through a discrete set ofpoint-value pairs a sampled dynamical system. Capturing the availableinformation with chain maps on Delaunay complexes, we use persistent homologyto quantify the evidence of recurrent behavior. We establish a sampling theoremto recover the eigenspace of the endomorphism on homology induced by theself-map. Using a combinatorial gradient flow arising from the discrete Morsetheory for \v{C}ech and Delaunay complexes, we construct a chain map totransform the problem from the natural but expensive \v{C}ech complexes to thecomputationally efficient Delaunay triangulations. The fast chain map algorithmhas applications beyond dynamical systems.

\v{C}ech-Delaunay gradient flow and homology inference for self-maps